Ideas that should be taken more seriously by Less Wrong:
Human beings are universal knowledge creators: they can create any knowledge that any other knowledge creator can create.
The only known tenable way of creating knowledge is by conjectures and refutations.
Induction is a myth.
Theories are either true or false: there is no such thing as the probability that a theory is true.
Confirmation does not make a theory more likely or better supported—the only role of confirmation is to provide a ready stock of criticisms of rival theories.
The most important knowledge is explanations.
There is no route to certain knowledge: we are all fallible.
We don’t need certain knowledge to progress: tentative, fallible, knowledge is just fine.
Gee, I wonder what philosopher of science you have been reading. :)
I would suggest that you read through the sequences with an open mind—particularly on your point #4. If you find it impossible to open your mind on that point, then open it to the possibility that the word “probability” can have two different meanings and that your point #4 only applies to one of them. If you find it impossible to open your mind to the possibility that a word might have an alternative meaning which you have not yet learned, then please go elsewhere.
Regarding Popper, it is not so much that he is wrong, as that he is obsolete. We think we have learned that set of lessons and moved on to the next set of problems.
If you have already begun reading the sequences, and were motivated to give us this dose of Popper because Eliezer’s naive realism got on your nerves, well …
All I can say is that it got on my nerves too, but if you keep reading you will find that EY is not nearly as epistemologically naive as it might seem in the early sequence postings.
No Popper is not obsolete and clearly the lessons of Popper have not been learnt by many: consider the people who have not yet understood that induction is a myth. Consider, also, the people who constantly misrepresent what Popper said like saying his philosophy is falsificationism or that he was a positivist or that he snuck induction in via the back door (you can find examples of these kind of mistakes discussed here). Popper’s ideas are in fact difficult for most people—they blow away the whole justificationist meta-context, a meta-context that permeates most people’s thinking. Understanding Popper requires that you take him seriously. David Deutsch did that and expanded on Popper’s ideas in a number of ways (you may be interested in a new book he has coming out called “The Beginning of Infinity”). He is another philosopher I follow closely. As is Elliot Temple (www.curi.us).
Thanks for the links and references. I will look into them. I urge you once more to work your way through the sequences. It appears you have something to teach us, but I doubt that you will be very successful until you have learned the local jargon, and become sufficiently familiar with our favorite examples to use them against us.
However, I have to say that I was a bit disconcerted by this:
consider the people who have not yet understood that induction is a myth.
Now if you told me that the standard definition of induction misrepresents the evidence-collection process, or that you know how to dissolve the problem of induction, well then I would be all ears. But when you say that “induction is a myth” I hear that as saying that everyone who has thought seriously on the topic, from Hume to the present, …,
well, you seem to be saying that all those smart people were as deluded as the medieval philosophers who worried about angels dancing on the heads of pins.
See the thing is, I would have to keep having to upvote such arrogance and stupidity, just so the comment to which I am responding doesn’t disappear. And I don’t want to do that.
You do realize that Hume held that induction cannot be logically justified? He noticed there is a “problem of induction”. That problem was exploded by Karl Popper. Have you read what he has to say and taken seriously his ideas? Have you read and taken seriously the ideas of philosophers like David Deutsch, David Miller, and Bill Bartley? They all agree with Popper that:
Induction, i.e. inference based on many observations, is a myth. It is neither a psychological fact, nor a fact of ordinary life, nor one of scientific procedure - Karl Popper (Conjectures & Refutations, p 70).
You do realize that Hume held that induction cannot be logically justified? He noticed there is a “problem of induction”.
Of course. That is why I mentioned him.
That problem was exploded by Karl Popper. Have you read what he has to say and taken seriously his ideas?
“Exploded”. My! What violent imagery. I usually prefer to see problems “dissolved”. Less metaphorical debris. And yes, I’ve read quite a bit of Popper, and admire much of it.
Have you read and taken seriously the ideas of philosophers like David Deutsch, David Miller, and Bill Bartley?
Nope, I haven’t.
They all agree with Popper that:
Induction, i.e. inference based on many observations, is a myth. It is neither a psychological fact, nor a fact of ordinary life, nor one of scientific procedure—Karl Popper (Conjectures & Refutations, p 70).
You know, when giving page citations in printed texts, you should specify the edition. My 1965 Harper Torchbook paperback edition does not show Popper saying that on p 70. But, no matter.
One of the few things I dislike about Popper is that he doesn’t seem to understand statistical inference. I mean, he is totally clueless on the subject. It is not just that he isn’t a Bayesian—it seems he doesn’t “get” Pearson and Fisher either. Well, no philosopher gets everything right. But if he really thinks that “inference based on many observations” cannot happen—not just that it is frequently done wrong, but rather that it is impossible—then all I can say is that this is not one of Sir Karl’s better moments.
And if what he means is simply that we cannot infer absolute general truths from repeated observations, then I have to call him a liar for suggesting that anyone else ever suggested that we could make such inferences.
But, since you have been recommending philosophers to me, let me recommend some to you. I. J. Good is fun. Richard Jeffrey is not bad either. E.T. Jaynes explains quite clearly how one makes inferences based on observations—one observation or many observations. You really ought to look at Jaynes before coming to this forum to lecture on epistemology.
Perhaps you should know I have published papers where I have used Bayes extensively. I am well familiar with the topic (edit: though this doesn’t make me any kind of infallible authority). I was once enthusiastic about Bayesian epistemology myself. I now see it as sterile. Popperian epistemology—especially as extended by David Deutsch—is where I see fertile ground.
Cool. But more to the point, have you published, or simply written, any papers in which you explain why you now see it as sterile? Or would you care to recommend something by Deutsch which reveals the problems with Bayesianism. Something that actually takes notice of our ideology and tries to refute it will be received here much more favorably than mere diffuse enthusiasm for Popper.
if he really thinks that “inference based on many observations” cannot happen—not just
that it is frequently done wrong, but rather that it is impossible—then all I can say is that
this is not one of Sir Karl’s better moments.
The quote is from 3rd ed. 1968. You say you have read Popper, then you should not be surprised by the quote. Your response above is just the argument from incredulity. Do you have a better criticism?
I’m not surprised by the quote. I just couldn’t find it. It apparently wasn’t in 2nd edition. But my 2nd edition index had many entries for “induction, myth of _” so I don’t doubt you at all that Popper actually said it.
I am incredulous because I know how to do inference based on a single observation, as well as inference based on many. And so does just about everyone who posts regularly at this site. It is called Bayesian inference, and is not really all that difficult. Even you could do it, if you were to simply set aside your prejudice that
Theories are either true or false: there is no such thing as the probability that a theory is true.
I have already provided references. You can find thousands more by Googling.
OK, tell me how you know in advance of having any theory what to observe?
BTW, please don’t assume things about me like asserting I hold prejudices. The philosophical position I come from is a full blown one. - it is no mere prejudice. Also, I’m quite willing to change my ideas if they are shown to be wrong.
Ok, I won’t assume that you believe, with Popper whom you quote, that inference based on many observations is impossible. I will instead assume that Popper is using the word “inference” very differently than it is used around here. And since you claim to be an ex-Bayesian, I will assume you know how the word is used here. Which makes your behavior up until now pretty inexplicable, but I will make no assumptions about the reasons for that.
Likewise, please do not assume that I believe that observation is neither theory-laden nor theory-directed. As it happens, I do not know in advance of a theory what to observe.
Of course, the natural thing for me to do now would be to challenge you to explain where theories come from in advance of observation. But why don’t we both just grow up?
If you have a cite for a careful piece of reasoning which will cause us to drop our Bayesian complaisancy and re-embrace Popper, please provide it and let us read the text in peace.
If you have a cite for a careful piece of reasoning which will cause us to drop our Bayesian complaisancy and re-embrace Popper, please provide it and let us read the text in peace.
It sounds like Scurfield’s “cite for a careful piece of reasoning” are the works of Karl Popper, which you are also familiar with. I don’t have time to read the works of Karl Popper, but I have plenty of time to read blog comments about them. I’ve found every single comment in this thread interesting. Why discourage it?
I think the problem is a communication gap—”Bayesian” can mean different things to different people; and my best guess is that Scurfield converted from Laplace’s degree-of-belief approach to probability. Around here, though, the dominant Bayesian paradigm is Jaynes’, which takes the critiques of Bayes from the 1920 through the 1970s into account and digs through them to the epistemological bedrock below pretty well. Unless Scurfield has something new to say about Jaynes’ interpretation, his critiques aren’t that interesting to people who already know both Popper and Jaynes.
That can’t actually be everyone here. And I hope no one is offended if I say that Scurfield seems to “know Popper” to a greater degree than any of the other participants in this thread. Why the scorn for the guy and the conversation?
He certainly knows Popper better than me. I scorn the conversation because it is not stimulating me—not causing me to consider ideas I have never considered before. I scorn the guy (scorn may be a bit too strong here, but just go with it) because so far he has mostly presented slogans, rather than arguments. (Admitedly, I haven’t presented arguments either, but that is because his slogans strike me as either truisms or word games.)
The only thing I gained from this encounter was the link to the Critical Rationalism web site, where can be found links to writings by Popper and others. The CR site itself is, …, well, not great. For example, check out the “What is CR?” page where CR is contrasted with two other possible approaches to philosophy. Please actually check it out before continuing.
so far he has mostly presented slogans, rather than arguments.
It occurs to me that one thing he could do which would be both interesting and useful would be to go through the sequences, adding comments critiquing Eliezer’s epistemology lessons from the viewpoint of Popper and/or CR. Who knows? I might frequently find myself agreeing with him.
Indeed, that’s why I am in favor of voting on old comments. Ideally, people can continue to leave criticisms on the sequences, and good ones will rise to the top over time.
because so far he has mostly presented slogans, rather than arguments.
Yes, I asked for clarification of the slogans and got more slogans, and asked for arguments supporting the claims and was given the claims again. I decided at that point to disengage.
Now weren’t those subtle strawmen? :)
Indeed—I hadn’t bothered to check out the site, but it seems to me that most of the discipline of Philosophy falls outside “CR”’s “three major schools”, and they’re pretending Popper invented philosophy. It’s really quite terrible.
If I may use another “slogan”: communication is difficult. And another: misunderstandings are common. When you asked for clarification I wasn’t sure what you wanted. I guessed and looks like I got it wrong. So you just withdraw? That’s very Un-Popperian.
It is a reasonable interpretation of the “three major schools” analysis down near the bottom of the “What is CR” page at the “Critical Rationalism” website. See if you can talk someone into cleaning up that bit of enthusiasm. As they say “It’s not helping”.
A better phrasing for that might have been “certain knowledge is a myth.” What cannot be logically justified is reasoning from particular observations to certainty in universal truths. You’re commenting as if you are unaware of the positions and arguments linked from my previous reply, and perhaps Where Recursive Justification Hits Bottom . You have intelligent things to say, but you’re not going to be taken seriously here if you’re not aware of the pre-existing intelligent responses to them popular enough to amount to public knowledge.
A better phrasing for that might have been “certain knowledge is a myth.” What cannot be logically justified
is reasoning from particular observations to certainty in universal truths.
No, that is not equivalent. Popper wrote that “inference based on many observations is a myth”. He is saying that we never reason from observations, never mind reasoning to certainty. In order to observe, you need theories. Without those, you cannot know what things you should observe or even make sense of any observation. Observation enables us to test theories, it never enables us to construct theories. Furthermore, Popper throws out the whole idea of justifying theories. We don’t need justification at all to progress. Judging from Where Recursive Justification Hits Bottom, this is something Eliezer has not fully taken on board (though I may be wrong). He sees the problem of the tu-quoque, but he still says [e]verything, without exception, needs justification. No, nothing can be justified. Knowledge advances not positively by justifying things but negatively by refuting things. Eliezer does see the importance of criticism, but my impression is that he doesn’t know Popper well enough.
“Previously, the most popular philosophy of science was probably Karl Popper’s falsificationism—this is the old philosophy that the Bayesian revolution is currently dethroning.”
“On the other hand, Popper’s idea that there is only falsification and no such thing as confirmation turns out to be incorrect. Bayes’ Theorem shows that falsification is very strong evidence compared to confirmation, but falsification is still probabilistic in nature; it is not governed by fundamentally different rules from confirmation, as Popper argued.”
Yudhowsky gets a lot wrong even in a few sentences:
Previously, the most popular philosophy of science was probably Karl Popper’s
falsificationism—this is the old philosophy that the Bayesian revolution is currently
dethroning.
First, Popper’s philosophy cannot be accurately described as falsificationism—that is just a component of it, and not the most important component. Popperian philosophy consists of many inter-related ideas and arguments. Yudhowsky makes an error that Popperian newbies make. One suspects from this that Yudhowsky is making himself out to be more familiar with Popper than he actually is. His claim to be dethroning Popper would then be dishonest as he does not have detailed knowledge of the rival position. Also he is wrong that Popper is popular: he isn’t. Furthermore, Popper is familiar with Bayesian epistemology and actually discusses it in his books. So calling Popper’s philosophy old and making out that Bayesian epistemology is new is wrong also.
Karl Popper’s idea that theories can be definitely falsified, but never definitely
confirmed, is yet another special case of the Bayesian rules;
Popper never said theories can be definitely falsified. He was a thoroughgoing fallibilist and viewed falsifications as fallible conjectures. Also he said that theories can never be confirmed at all, not that they can be partially or probabilistically confirmed, which the above sentence suggests he said. Saying falsification is a special case of the Bayesian rules also doesn’t make sense: falsification is anti-induction whereas Bayesian epistemology is pro-induction.
science itself is a special case of Bayes’ Theorem; experimental evidence is Bayesian evidence.
Science revolves around explanation and criticism. Most scientific ideas never get to the point of testing (which is a form of criticism), they are rejected via criticism alone. And they are rejected because they are bad explanations. Why is the emphasis in the quote solely on evidence? If science is a special case of Bayes, shouldn’t Bayes have something to say about explanation and criticism? Do you assign probabilities to criticism? That seems silly. Explanations and criticism enable us to understand things and to see why they might be true or false. Trying to reduce things to probabilities is to completely ignore the substance of explanations and criticisms. Instead of trying to get a probability that something is true, you should look for criticisms. You accept as tentatively true anything that is currently unproblematic and reject as tentatively false anything that is currently problematic. It’s a boolean decision: problematic or unproblematic.
Both bayesian induction (as we currently know it) and Popper fail my test for a complete epistemology.
The test is simple. Can I use the description of the formalism to program a real computer to do science? And it should, in theory, be able to bootstrap itself from no knowledge of science to our level.
Human beings don’t actually seem to have utility functions, all they really have are “preferences” i.e. a method for choosing between alternatives. But von Neumann and Morgenstern showed that under some conditions this is the same as having a utility function.
Now Scurfield is saying that human beings, even smart ones like scientists, don’t have prior probability distributions, all they really have is a database of claims and criticisms of those claims. Is there any result analogous to von Neumann-Morgenstern that says this is the same thing as having a prior, under conditions?
Yes. The question has been addressed repeatedly by a variety of people. John Maynard Keynes may have been the first. Notable formulations since his include de Finetti, Savage, and Jeffrey’s online book.
Discovering subjective probabilities is usually done in conjunction with discovering utilities by revealed preferences because much of the machinery (choices between alternatives, lotteries) is shared between the two problems. People like Jaynes who want a pure epistemology uncontaminated by crass utility considerations have to demand that their “test subjects” adhere to some fairly hard-to-justify consistency rules. But people like de Finetti don’t impose arbitrary consistency, instead they prove that inconsistent probability assignments lose money to clever gamblers who construct “Dutch books”.
I’m not even familiar with Halpern’s work. The only serious criticism I have seen regarding the usual consistency rules for subjective probabilities dealt with the “sure thing rule”. I didn’t find it particularly convincing.
No, I have no trouble justifying a mathematical argument in favor of this kind of consistency. But not everyone else is all that convinced by mathematics. Their attention can be grabbed, however, by the danger of being taken to the cleaners by Dutch book professional bookies.
One of these days, I will get around to producing a posting on probability, developing it from what I call the “surprisal” of a proposition—the amount, on a scale from zero to positive infinity, by which you would be surprised upon learning that a proposition is true.
Prob(X) = 2^(-Surp(X)).
Surp(coin flip yields heads)= 1 bit.
Surp(A) + Surp(B|A) = Surp(A&B)
That last formula strikes me as particularly easy to justify (surprisals are additive). Given that and the first formula, you can easily derive Bayes law. The middle formula simply fixes the scale for surprisals. I suppose we also need a rule that Surp(True)=0
Cool! Saves me the trouble of writing that posting. :)
Absurdity is probably a better name for the concept. Except that it sounds objective, whereas amount of surprise more obviously depends on who is being surprised.
I think that the contribution that Bayesian methodology makes toward good criticism of a scientific hypothesis is that to “do the math”, you need to be able to compute P(E|H). If H is a bad explanation, you will notice this when you try to determine (before you see E) how you would go about computing P(E|H). Alternately, you discover it when you try to imagine some E such that P(E|H) is different from P(E|not H).
No, you don’t assign probabilities to criticisms, as such. But I do think that every atomic criticism of a hypothesis H contains at its heart a conditional proposition of the form (E|H) or else a likelihood odds ratio P(E|H)/P(E|not H) together with a challenge, “So how would you go about calculating that?”
Incidentally, you also ought to look at some of the earlier postings where EY was, in effect, using naive Bayes classifiers to classify (i.e. create ontologies), rather than using Bayes’s theorem to evaluate hypotheses that predict. Also take a look at Pearl’s book to get a modern Bayesian view of what explanation is all about.
I like this point a lot. But it seems very convenient and sensible to say that some things are more problematic than others. And at least for certain kinds of claims it’s possible to quantify how problematic they are with numbers. This leads one (me at least) to want a formalism—for handling beliefs—that involves numbers, and Bayesianism is a good one.
What’s the conjectures-and-refutations way of handling claims like “it’s going to snow in February”? Do you think it’s meaningless or useless to attach a probability to that claim?
There is no problem with theories that make probabilistic predictions. But getting a probabilistic prediction is not tantamount to assigning a probability to the theory that made the prediction.
True. But you seem to be assuming that a “theory” has to be a universal law of nature. You are too attached to physics. But in other sciences, you can have a theory which is quite explanatory, but is not in any sense a “law”, but rather it is an event. Examples:
the theory that the moon was formed by a collision between the earth and a Mars-sized planetesimal.
the theory that modern man originated in Africa within the past 200,000 years and that the Homo erectus population outside of Africa did not contribute to our ancestry.
the theory that Napolean was poisoned with arsenic in St. Helena.
the “aquatic ape theory”
the endosymbiotic theory of the origin of mitochondria
the theory that the Chinese discovered America in 1421.
Probabilities can be assigned to these theories.
And even for universal theories, you can talk about the relative odds of competing theories being correct—say between a supersymetric GUT based on E6 and one based on E8. (Notice, I said “talk about the odds”, not “calculate them”) And you can definitely calculate how much one particular experimental result shifts those odds.
As you pointed out earlier, we have two ostensibly different ways of investigating the theory that the Chinese discovered America in 1421: the Popperian way, in which this theory and alternatives to it are criticized. And the Bayesian way, in which those criticisms are broken down into atomic criticisms, and likelihood ratios are attached and multiplied.
I’ve seen plenty of rigorous Popperian discussions but not very many very rigorous—or even modestly rigorous—Bayesian discussions, even on this website. One piece of evidence for the China-discovered-America theory is some business about old Chinese maps. How does a Bayesian go about estimating the likelihood ratio P(China discovered America | old maps) / P(China discovered America | no old maps)?
I think you want to ask about P(maps|discover) / P(no maps|discover). Unless both wikipedia and my intuition are wrong.
Does catching you in this error relieve me of the responsibility of answering the question?
I hope so. Because I would want to instead argue using something like P(maps|discover) vs P(maps|not discover). That doesn’t take you all the way to P(discover), but it does at least give you a way to assess the evidential weight of the map evidence.
Here’s another personal example of Bayesianism in action. Do you have a sense of how much you updated by? P(Richard Dawkins praises Steven Pinker | EP is bunk)/ P(Richard Dawkins praises Steven Pinker | EP is not bunk) is .5? .999? Any idea?
P(“Sewing Machine” is a nym) = 1.0 P(Sewing Machine has been disingenuous) = 0.5 and rising P(Dawkins praises Pinker|EP is not bunk) is ill defined because P(EP is not bunk) = ~0 but I have updated P(Dawkins believes EP is not bunk) to at least 0.5
the theory that the moon was formed by a collision between the earth and a Mars-sized planetesimal.
The reason we would accept this theory as true is because it has survived criticism as an explanation while its rivals have not. If another rival theory is still in contention by also having survived criticism then there is a problem and this problem is not going to be resolved by computing, somehow, probabilities that the theories are true. To solve the problem you are going to have to come up with better criticisms or, possibly, alternative theories.
One difference between theories and events is that the counterfactuals of an event exist (in the multiverse). So it makes sense to talk about the probability of an event: the counterfactual events are real and occur to a greater or lesser measure in the multiverse. A false theory, by contrast, is simply false, it has no connection to reality. How do you assign anything other than an arbitrary probability to something that simply cannot be? Fortunately we don’t have to: we have Popperian epistemology.
A false theory, by contrast, is simply false, it has no connection to reality.
How do you assign anything other than an arbitrary probability to something
that simply cannot be?
Intrade knows that one! At the Higgs boson bet over 200 people think they know how to assign a probability to the issue.
1: By using the counterfactuals in the Tegmark Level IV multiverse.
2: By giving it a probability of 0. If T is falsified, that means P(D|T)=0 - we obtained data that T claims is impossible. In this case, Bayes’ theorem sets P(T|D)=0. Bayesianism includes all correct thinking tools, including Popperian epistemology.
But is P(D|T) really 0? We could have made a mistake and not recorded the correct data. Certainly scientists in the past have done so, and thought that they falsified theories that they didn’t falsify. In this case, P(D|T) is very small but nonzero, and so is P(T|D) (unless p(D|~T) is also very small.)
3: You cannot avoid giving a probability. Because of Cox’s theorem, which says we must use probability theory to reason about uncertainty (although I must confess that the assumption that we must use a single real number to reason is rather strong.)
Counterfactuals of the sort of theory that Perplexed describe do exist in a Quantum Multiverse; why not assign them probabilities?
Anyway, why would you want to make your entire epistemology contingent on a particular theory of physics? It sounds like the CR would collapse if Copenhagen turned out to be correct.
Yes, counterfactuals do exist in some of Perplexed examples. There are alternative universes where Earth does not have a moon, or it does have a moon but it was not formed by a collision with a planetesimal. However, in this universe, the one we are observing now, the moon either was or it was not formed in such a way. There is no middle ground. Finding evidence consistent with the theory does not make the theory truer or more likely—what the evidence does is supply us with criticisms of rival theories.
CR is not independent from physics, nor can it be. The laws of physics entail the existence of universal computers and of universal knowledge creators. As David Deutsch has shown, there are deep connections between multiversal quantum physics and the theory of information. If Cophenhagen should turn out to be true it would impact on many things and not the least of which would be CR.
Events have probabilities. Theories don’t. Given some theory of meteorology, you can predict the probability it will snow in Feb. But you can’t say the probability that that theory of meteorology is true.
Did you read the context? Someone asked the Popperian view on giving a probability of future weather. So I answered that. What exactly do you think the context is?
The scientific method as a special case of Bayes theorem and whether the not directly experimental aspects can be mapped on some part of Bayesian reasoning. Now that you pointed it out I can see that you were only referring to the narrow sub-point in the great-gandparent and not the wider context, but it looked to me like you were also arguing that since theories don’t have (frequentlialist) probabilities Popperian reasoning about them couldn’t map to probabilistically framed Bayesian reasoning. Looking at the votes it seems I wasn’t alone in that (mis-)reading.
A criticism can have many components, some of which are correct and some of which are incorrect. Breaking a criticism down into its components can be difficult/problematic.
Edit: The way I put that sounds stupid. Let me try again: occasionally, a pair of math papers are released, one purports to prove a conjecture, and one purports to disprove it. The authors then criticize each others papers (let’s say). Would you really characterize the task of assigning probabilities in this situation as “unproblematic”?
Maybe you would do that. I would instead bog down in a discussion of whether the criticism was a nitpick or a “real” criticism. But I would be interested to see what odds ratio you come up with for this criticism being correct.
And in the math papers example, how exactly are you going to do that? Presumably you are going to go through the papers and the criticisms in detail and evaluate the content. And when you do that you are going to think of reasons why one is right and the other wrong. And then probabilities become irrelevent. It’s your understanding of the content that will enable you to choose.
I don’t think anyone is falling into this trap. It sounds like the Popperian version is replacing “true” and “false” by “tentatively true” and “tentatively false.”
Theories are either true or false. The word “tentative” is there as an expression of fallibility. We cannot know if a theory is in fact true: it may contain problems that we do not yet know about. All knowledge is tentative. The word is not intended as a synonym for probability or to convey anything about probabilities.
Observers can put probabilities on the truth of theories. They can do it—and will do it—if you ask them to set odds and prepare to receive bets. Quantifying uncertainty allows it to be measured and processed.
It is true that knowledge is fallible—but some knowledge is more fallible than others—and if you can’t measure degrees of uncertainty, you will never develop a quantitative treatment of the subject. Philosophers of science realised this long ago—and developed a useful framework for quantifying uncertainty.
Observers can put probabilities on the truth of theories. They can do it—and will do it—if you ask them to set odds and prepare to receive bets. Quantifying uncertainty allows it to be measured and processed.
Scurfield missed his chance here. He should have asked when it becomes the case that those bets must be paid off, and offered the services of a Popper adept to make that kind of decision. Of course, the Popperite doesn’t rule that one theory is true, he rules that the other theory is refuted.
Short time limits don’t mean that agents can’t meaningfully assign probabilities to the truth of scientific theories—they just decrease the chances of the theories being proven wrong within the time limit a bit.
What is a time limit? Do actual bets on this sort of thing in Britain stipulate a time limit? As a Yank, I have no idea how betting ‘markets’ like this like this actually work.
Prediction markets/betting markets like Intrade or Betfair pretty universally set time limits on their bets. (Browse through Intrade sometime.) This does sometimes require changing the bet/prediction though—from ‘the Higgs boson will be found’ to ‘the Higgs boson will be found by 2020’. Not that this is a bad thing, mind you.
Not really. To the extent that we limit attention to theories of the form:
Always(Everywhere(Forall x (P(x)) ) )
we Bayesians can never “cash in” on a bet that the theory is true—at least not using empirical evidence. All we can do is to continue trying to falsify the theory by experiments at more times, at more places, and for more values of x. As Popper prescribes. Our probabilities that the theory is true grow higher and higher, but they grow more and more slowly, and they can never reach unity.
However, both Bayesians and Popper fans can become pretty certain that such a theory is false—even without checking everywhere, everywhen, and forall x. Popper does not have a monopoly on refutations. Or conjectures either, for that matter.
There are no degrees of fallibility. We are simply fallible: that’s it. You have no way of knowing if a currently unproblematic theory is wrong, no matter how obvious the theory may seem, it may end up being spectacularly wrong.
I agree that we can quantity the uncertainty of events and that this is useful.
But theories are a different kettle of fish. Popperian epistemology tells us that we don’t need to know anything about the uncertainty of theories for knowledge to grow. Hence, one does not need to quantify the uncertainty of theories in order to write a knowledge creating computer program.
The thing is, we have a beautiful theory of uncertainty that deals with uncertainty over events, uncertainty about hypotheses—uncertainty about all beliefs, in fact—and it works just great.
Sure, we could go back to the dark days before Bayes, and struggle on with a boolean conception of certainty, and it probably wouldn’t be so bad that it would actually prevent knowledge from accumulating...
...but what possible reason would motivate us to take such a retrograde step?
I mean: how do you model equiprobable competing theories in such a framework.
How do you model induction? How do you model confirmation?
The answer seems to be that you don’t—you just deny the very existence of these phenomena!
I hope you can see how that is not a step forwards, from our point of view.
The thing is, we have a beautiful theory of uncertainty that deals with uncertainty over events, uncertainty about hypotheses—uncertainty about all beliefs, in fact—and it works just great.
Really? No exaggeration?
Sure, we could go back to the dark days before Bayes, and struggle on with a
boolean conception of certainty, and it probably wouldn’t be so bad that it would
acutually prevent knowledge from growing...
Popperian epistemology was created in the 20th century, after Bayes, so what do you mean by dark days. Certainty is not the lynchpin of knowledge creation and in fact has nothing to do with it. You completely devalue the role of explanations and criticisms. Who cares about probability when you have a good explanation that has withstood criticism?
...but what possible reason would motivate us to take such a retrograde step?
Popperian philosophy is about problem-solving, explanation and criticism and these things are not in any way retrograde. Bayesian epistemology is rooted in the
old philosophy of justificationism, a philosophy that Popperian epistemology overturned.
I mean: how do you model equiprobable competing theories in such a framework.
What you do is attempt further criticism of these theories and if that doesn’t progress come up with a meta-theory of what to do giving that you have two good candidate theories. It is always possible to come up with such a meta-theory.
How do you model induction? How do you model confirmation?
The answer seems to be that you don’t—you just deny the very existence of these
phenomena!
Popper and others have given explanations—they don’t just deny without reason. Are you familiar with their arguments? There are real problems with the very concepts of induction and confirmation, problems that you seem not to have appreciated.
I hope you can see how that is not a step forwards, from our point of view.
Popper showed that knowledge grows perfectly fine without concepts of uncertainty, induction, and confirmation. Yes, it is counter-intuitive—that is why most people do not get Popper.
Popper and others have given explanations—they don’t just deny without reason. Are you familiar with their arguments? There are real problems with the very concepts of induction and confirmation, problems that you seem not to have appreciated.
Maybe you should present them, rather than playing ‘I can assert my philosopher is great more than you can’.
The author of that link thinks Popper is falsification. I have already explained why that view of Popper is wrong. Have you been paying attention? And does the author of the link realize that the leading exponent of the multiverse is David Deutsch, who also happens to be the best living Popperian?
“Popper’s great and tireless efforts to expunge the word induction from scientific and philosophical discourse has utterly failed. Except for a small but noisy group of British Popperians, induction is just too firmly embedded in the way philosophers of science and even ordinary people talk and think.
Confirming instances underlie our beliefs that the Sun will rise tomorrow, that dropped objects will fall, that water will freeze and boil, and a million other events. It is hard to think of another philosophical battle so decisively lost.”—M.Gardner.
Gardner can be taken as seriously on Popper as he can on the MWI, i.e, not at all.
BTW, the sun does not rise in Murmansk in the middle of winter, live flies that are dropped do not fall, and water can be prevented from freezing in my car radiator by adding anti-freeze.
Carnap had a major influence on me. He persuaded me that all metaphysical questions are “meaningless” in the sense that they cannot be answered empirically or by reason. They can be defended only on emotive grounds. Carnap was an atheist, but I managed to retain my youthful theism in the form of what is called “fideism.” I like to call it “theological positivism,” a play on Carnap’s “logical positivism.
As far as we can tell, universes are not as plentiful as even two blackberries. Surely the conjecture that there is just one universe and its Creator is infinitely simpler and easier to believe than that there are countless billions upon billions of worlds, constantly increasing in number and created by nobody. I can only marvel at the low state to which today’s philosophy of science has fallen
So why the downvote? Gardner doesn’t understand quantum physics and he doesn’t understand epistemology.
I still recommend
Subjectively Objective, but I’m no longer confident that your inferential distance from the coverage there is small enough. Perplexed’s recommendation to read all the way through the sequences, or—even better—ET Jaynes’ Probability Theory: The Logic of Science—may be necessary. As he’s said, Critical Rationalism was an important step in the philosophy of science—but the field has moved beyond that to a rigorous, mathematically precise model of the amount of belief any rational agent must hold given identical priors and the same evidence—Popper’s Vs(a)=CT(a)-CTf(a) is not quantitative in this way.
That wasn’t intended to convince you; if you truly wish to subject your conjecture to criticism a contemplative reading of Jaynes is necessary. If you do happen to find Jaynes convincing, all is not lost—we still like Tarski here.
Induction, i.e. inference based on many observations, is a myth. It is
neither a psychological fact, nor a fact of ordinary life, nor one of scientific
procedure—Karl Popper (Conjectures & Refutations, p 70).
Human beings are universal knowledge creators: they can create any knowledge that any other knowledge creator can create.
In what sense do you mean this exactly, and what evidence for it do you have? I’ve spoken to people like Elliot, but all they said was things like ‘humans can function as a Turing Machine by laboureously manipulating symbols’. Which is nice, but not really relevant to anything in real-time.
On a more general note, you should probably try to be a little clearer: ‘conjectures and refutations’ doesn’t really pick out any particular strategy from strategy-space, and neither does the phrase ‘explanation’ pick out anything in particular. Additionally, ‘induction’ is sufficiently different from what people normally think of as myths that it could do with some elaboration.
Human beings are universal knowledge creators: they can create any knowledge that any other knowledge creator
can create.
In what sense do you mean this exactly,
Another way of saying it is that human beings can solve any problem that can be solved. Does that help?
and what evidence for it do you have?
Careful here—as I mentioned above, evidence never supports a theory, it just provides a ready stock of criticisms of rival theories. Let me give you an argument: If you hold that human beings are not universal knowledge creators, then you are saying that human knowledge creation processes are limited in some way, that there is some knowledge we cannot create. You are saying that humans can create a whole bunch of knowledge but whole realms of other knowledge are off limits to us. How does that work? Knowledge enables us to expand our abilities and that in turn enables us to create new knowledge and so on. Whatever this knowledge we can’t create is, it would have to be walled off from all this other expanding knowledge in a rather special way. How do you build a knowledge creation machine that only has the capability to create some knowledge? That would seem much much more difficult than creating a fully universal machine.
I’ve spoken to people like Elliot, but all they said was things like ‘humans > can function as a Turing Machine by
laboureously manipulating symbols’. > Which is nice, but not really relevant to anything in real-time.
I don’t know what point Elliot was answering here, but I guess he is saying that humans are universal Turing Machines and illustrating that. He is saying that humans are universal in the sense that they can compute anything that can be computed. That is a different notion of universality to the one under discussion here (though there is a connection between the two types of universality). Elliot agrees that humans are universal knowledge creators and has written a lot about it (see, for example, his posts on The Fabric of Reality list).
On a more general note, you should probably try to be a little clearer: ‘conjectures and refutations’ doesn’t really pick
out any particular strategy from strategy-space,
‘Conjectures and refutations’ is an evolutionary process. The general methodology (or strategy, if you prefer) is: When faced with a problem try to come up with conjectural explanations to solve the problem and then criticise them until you find one (and only one) that cannot be knocked down by any known criticism. Take that as your tentative solution. I guess what you are looking for is an explanation of how human conjecture engines work? That is an unsolved problem. We do know some things, eg: no induction is involved.
and neither does the phrase ‘explanation’ pick out anything in particular.
Explanations are valuable: they help you understand something. Are you looking for an explanation of how we generate “explanations”? Again, unsolved problem.
Additionally, ‘induction’ is sufficiently different from what people normally think of as myths that it could do with some
elaboration.
It’s not really different. It’s something that people believe is true that in fact isn’t. Hume was the first to realize that there was a “problem of induction” and philosophers have for years and years been trying to justify induction. It took Karl Popper to realize that induction isn’t actually how we create knowledge at all: induction is a myth.
Similarly, some of these issues we do take seriously; we know we’re fallible,
Yes, you are called “Less Wrong” after all! I was off-beam with that.
and it sounds like you don’t know what we mean by probability.
Actually, I am quite familiar with the Bayesian conception of probability. I just don’t think probability has a role in the realm of epistemology. Evidence does not make a theory more probable, not even from a subjective point of view. What evidence does, as I have said, is provide a stock of criticisms against rival theories. Also, evidence only goes so far: what really matters is how theories stand up to criticism as explanations. Evidence plays a role in that. I am quite happy to talk about the probability of events in the world, but events are different from explanatory theories. Apples and oranges.
Another way of saying it is that human beings can solve any problem that can be solved. Does that help?
What about the problem of building pyramids on alpha-centuri by 2012? We can’t, but aliens living there could.
More pressingly though, I don’t see why this is important. Have we been basing our arguments on an assumption that there are problems we can’t solve? Is there any evidence we can solve all problems without access to arbitrarily large amounts of computational power? Something like AIXI can solve pretty much anything, but not relevantly.
That would seem much much more difficult than creating a fully universal machine.
How about a neural network that can’t learn XOR?
When faced with a problem try to come up with conjectural explanations to solve the problem and then criticise them until you find one (and only one) that cannot be knocked down by any known criticism.
The manner in which explanations are knocked down seems under-specified, if you’re not doing Bayesian updating.
Are you looking for an explanation of how we generate “explanations”? Again, unsolved problem.
Nope, I just don’t know what in particular you mean by ‘explanation’. I know what the word means in general, but not your specific conception.
I just don’t think probability has a role in the realm of epistemology.
Well, that’s different from there being no such thing as a probability that a theory is true: your initial assertion implied that the concept wasn’t well defined, whereas now you just mean it’s irrelevant. Either way, you should probably produce some actual arguments against Jaynes’s conception of probability.
Meta: You want to reply directly to a post, not its descendants, or the other person won’t get a notification. I only saw your post via the Recent Posts list.
Also, it’s no good telling people that they can’t use evidence to support their position because it contradicts your theory when the other people haven’t been convinced of your theory.
The manner in which explanations are knocked down seems under-specified, if you’re not
doing Bayesian updating.
Criticism enables us to see flaws in explanations. What is under-specified about finding a flaw?
In your way, you need to come up with criticisms and also with probabilities associated with those criticisms. Criticisms of real world theories can be involved and complex. Isn’t it enough to expose a flaw in an explanatory theory? Must one also go to the trouble of calculating probabilities—a task that is surely fraught with difficulty for any realistic idea of criticism? You’re adding a huge amount of auxilliary theory and your evaluation is then also dependent on the truth of all this auxilliary theory.
I just don’t know what in particular you mean by ‘explanation’. I know what the word means
in general, but not your specific conception.
You don’t seem to be actually saying very much then; is LW really short on explanations, in the conventional sense? Explanation seems well evidenced by the last couple of top level posts. Similarly, do we really fail to criticise one another? A large number of the comments seem to be criticisms. If you’re essentially criticising us for not having learn rationality 101 - the sort of rationality you learn as a child of 12, arguing against god—then obviously it would be a problem if we didn’t bare in mind the stuff. But without providing evidence that we succumb to these faults, it’s hard to see what the problem is.
Your other points, however, are substantive. If humans could solve any problem, or it was impossible to design an agent which could learn some but not all things, or confirmation didn’t increase subjective plausibility, these would be important claims.
Actually, I am quite familiar with the Bayesian conception of probability.
I just don’t think probability has a role in the realm of epistemology. Evidence
does not make a theory more probable, not even from a subjective point of view.
Of course evidence makes theories more probable:
Imagine you have two large opaque bags full of beans, one 50% black beans and 50% white beans and the other full of white beans. The bags are well shaken, you are given one bag at random. You take out 20 beans—and they are all white.
That is clearly evidence that confirms the hypothesis that you have the bag full of white beans. If you had the “mixed” bag, that would only happen one time in a million.
Notice that the counterfactual event is possible (that you have the mixed bag). And even if you hold the bag full of white beans, the counterfactual event that you hold the mixed beans does occur elsewhere in the multiverse. This is what distinguishes events from theories. A false theory never obtains anywhere: it is simply false. So a theory being true or false is not at all like the situation with counterfactual events. You cannot assign anything objective to a false theory.
The actual theory you hold in your example is approximately the following: I have made a random selection from a bag and I know that I have been given one of two bags: one 50% black beans and 50% white beans and the other full of white beans and: I have been honestly informed about the setup, am not being tricked, no mistakes have been made etc. This theory predicts that if I take 20 white beans out of the bag, then the chance of that would be one in a million if I had the mixed bag. Do you understand? The real situation is that you have a theory that is making probabilistic predictions about events and, as I have said several times, I have no problem with probabilistic predictions of theories about events.
Firstly, this seems like a step forwards to me. You seem to agree that induction and confirmation are fine 90% of the time. You seem to agree that these ideas work in practice—and are useful—including in some realms of knowledge—such as knowledge relating to which bag is in front of you in the above example. This puts your anti-induction and anti-confirmation statements into a rather different light, IMO.
Confirmation theory has nothing to do with multiverses. It applies equally well for agents in single deterministic universes—such as can be modelled by cellular automata. So, reasoning that depends on the details of multiverse theories is broken from the outset. Imagine evidence for wavefunction collapse was found.
Not terribly likely—but it could happen—and you don’t want your whole theory of epistemology to break if it does!
Treating uncertainty about theories and uncertainty about events differently is a philosophical mistake. There is absolutely no reason to do it—and it gets people into all kinds of muddles.
We have a beautiful theory of subjective uncertainty that applies equally well to uncertainty about any belief—whether it refers to events, or scientific theories. You can’t really tease these categories apart anyway—since many events are contingent upon the truth of scientific theories—e.g. Higgs boson observations. Events are how physical law is known to us.
Instead of using one theory for hypotheses about events and another for hypotheses about universal laws you should—according to Occam’s razor—be treating them in the same way—and be using the same underlying general theory that covers all uncertain knowledge—namely the laws of subjective probability.
Tim—In the example we have been discussing, no confirmation of the actual theory (the one I gave in approximate outline) happens. The actual theory makes probabilistic predictions about events (it also makes non-probabilistic predictions) and tells you how to bet. Getting 20 white beans doesn’t make the actual theory any more probable—the probability was a prediction of the theory. Note also that a theory that you are being tricked might recommend that you choose the mixed bag when you get 20 white beans. Lots of theories are consistent with the evidence. What you need to look for is things to refute the possible theories. If you are concerned with confirmation, then the con man wins.
So I am not agreeing that induction and confirmation are fine any percentage of the time (how did you get that 90% figure?). When you consider the actual possible theories of the example, all that is happening is that you have explanatory theories that make predictions, some probabilistic, and that tell you how to bet. The theories are not being induced from evidence and no confirmation takes place.
You haven’t explained how we assign objective probabilities to theories that are false in all worlds.
What you need to look for is things to refute the possible theories. If you are concerned with confirmation, then the con man wins.
What you’re talking about here is a strategy for avoiding bias which Bayesians also use. It is not a fundamental feature of any particular epistemology.
You don’t seem to address the idea that multiverse theories are an irrelevance—and that in a single deterministic automaton, things work just the same way.
Indeed, scientists don’t even know which (If any) laws of physics are true everywhere, and which depend on the world you are in.
You don’t seem to address the idea that we have a nice general theory that covers all kinds of uncertainty, and that no extra theory to deal with uncertainty about scientific hypotheses is needed.
If you don’t class hypotheses about events as being “theories”, then I think you need to look at:
Also, your challenge doesn’t seem to make much sense. The things people assign probabilities to are things they are uncertain about. If you tell me a theory is wrong, it gets assigned a low probability. The interesting cases are ones where we don’t yet know the answer—like the clay theory of the origin of life, the orbital inclination theory of glacial cycles—and so on.
Distinguishing between scientific theories and events in the way that you do apparently makes little sense. Events depend on scientific theories. Scientific theories predict events. Every test of a scientific theory is an event. Observing the perihelion precession of Mercury was an event. The observation of the deflection of light by the Sun during an eclipse was an event. If you have probabilities about events which are tests of scientific theories, then you automatically wind up with probabilities about the theories that depend on their outcome.
Basically agents have probabilities about all their beliefs. That is Bayes 101. If an agent claims not to have a probability about some belief, you can usually set up a bet which reveals what they actually think about the subject. Matters of fundamental physics are not different from “what type of beans are in a bag”—in that respect.
Scientific theories predict events. Every test of a scientific theory is an event.
Observing the perihelion precession of Mercury was an event. The observation of
the deflection of light by the Sun during an eclipse was an event.
Yes, scientific theories predict events. So there is a distinction between events and theories right? If the event is observed to occur, all that happens is that rival theories that do not predict the event are refuted. The theory that predicted the event is not made truer (it already is either true or false). And there are always an infinite number of other theories that predict the same event. So observing the event doesn’t allow you to distinguish among those theories.
In the bean bag example you seem to think that the rival theories are “the bag I am holding is mixed” and “the bag I am holding is all white”. But what you actually have is a single theory that makes predictions about these two possible events. That theory says you have a one-in-a-million chance of holding the mixed bag.
Matters of fundamental physics are not different from “what type of beans are in a
bag”
No, General Relativity being true or false is not like holding a bag of white beans or holding a bag of mixed beans. The latter are events that can and do obtain: They happen. But GR is not true in some universes and false in others. It is either true or false. Everywhere. Furthermore, we accept GR not because it is judged most likely but because it is the best explanation we have.
Popperians claim that we don’t need any theory of uncertainty to explain how knowledge grows: uncertainty is irrelevant. That is an interesting claim don’t you think? And if you care about the future of humanity, it is a claim that you should take seriously and try to understand.
If you are still confused about my position, why don’t you try posting some questions on one of the following lists:
It might be useful for other Popperians to explain the position—perhaps I am being unclear in some way.
Edit: Just because people might be willing to place bets is no argument that the epistemological point I am making is wrong. What makes those people infallible authorities on epistemology? Also, if I accept a bet from someone that a universal theory is true, would I ever have to pay out?
In the bean bag example you seem to think that the rival theories are “the bag I am holding is mixed” and “the bag I am holding is all white”. But what you actually have is a single theory that makes predictions about these two possible events. That theory says you have a one-in-a-million chance of holding the mixed bag.
That’s a really powerful general argument against Bayesianism that I hadn’t considered before: any prior (edit: I should have said “prior information”) necessarily constitutes a hypothesis in which you have confidence 1.
I don’t think that statement makes sense; you seem to be mixing levels—the prior is a distribution over how the world could actually be, not over other distributions. It shouldn’t make sense to speak of your prior’s confidence in itself.
You have an explanatory theory that makes predictions about the events, but it is not the only possible explanatory theory. If someone offers to play the bean bag game with you on the street, then things might not be as they seem and your theory would be no good as an explanation of how to bet. Science is like that—what is actually going on might not be what you think, so you look for flaws and realize that one’s confidence is no guide to the truth.
If your confidence in your prior were 1, you would never be able to update it. But, it is true that if your prior distribution of probabilities over various hypotheses assigns 0 or 1 probability to a group of hypotheses, you will never be able to accrue enough evidence to change that. This is not a weakness of Bayesianism, because there is no other method of reasoning which will allow you to end up on a conclusion which you at no point considered as a possibility.
Did you read the quoted text? Inability to update is the whole point of my concern; but it in no way implies that my confidence in a particular outcome will never change.
Perhaps you’re confusing probabilities for priors. (edit: I was misusing my terms: I meant “prior probabilities” and “prior information” respectively.)
I think that the problem is that EY has introduced non-standard terminology here. Worse, he blames it on Jaynes, who makes no such mistake. I just looked it up.
There are two concepts here which must not be confused.
a priori information, aka prior information, aka background information
prior probabilities, aka priors (by everyone except EY. Jaynes dislikes this but acquiesces).
Prior information does indeed constitute a hypothesis in which you have complete confidence. I agree this is something of a weakness—a weakness which is recognized implicitly in such folklore as “Cromwell’s rule” Prior information cannot be updated.
Prior probabilities (frequently known simply as priors) can be updated. In a sense, being updated is their whole purpose in life.
You are welcome. Unfortunately, I was wrong. Or at least incomplete.
I misinterpreted what EY was saying in the posting you cited. He was not, as I mistakenly assumed, saying that prior probabilities should not be called priors. He was instead talking about a third kind of entity which should not be confused with either of the other two.
Prior distributions over hypotheses, which Eliezer wishes to call simply “priors”
But there is not a confusion with referring to both prior probabilities and prior distributions as simply priors because a prior probability is simply a special case of a prior distribution. A probability is simply a distribution over a set of two competing hypotheses—only one of which can be true.
Bayes theorem in its usual form applies only to simple prior probabilities. It tells you how to update the probability. In order to update a prior distribution, you effectively need to use Bayes’s theorem multiple times—once for each hypothesis in your set of hypotheses.
So what is that 1⁄2 number which Eliezer says is definitely not a prior? It is none of the above three things. It is something harder to describe. A statistic over a distribution. I am not even going to try to explain what that means.
Sorry for any confusion I may have created. And thx to Sniffnoy and timtyler for calling my attention to my mistake.
This can easily be “flattened” into a single, more complex, probability distribution:
25% draw white bean from mixed bag.
25% draw black bean from mixed bag.
50% draw white bean from unmixed bag.
If we wish to consider multiple draws, we can again flatten the total event into a single distribution:
1⁄8 mixed bag, black and black
1⁄8 mixed bag, black and white
1⁄8 mixed bag, white and black
1⁄8 mixed bag, white and white
1⁄2 unmixed bag, white and white
Translating the “what is that number” question into this situation, we can ask: what do we mean when we say that we are 5⁄8 sure that we will draw two white beans? I would say that it is a confidence; the “event” that has 5⁄8 probability is a partial event, a lossy description of the total event.
I’m not convinced that there’s a meaningful difference between prior distributions and prior probabilities.
There isn’t when you have only two competing hypotheses. Add a third hypothesis and
you really do have to work with distributions. Chapter 4 of Jaynes explains this wonderfully. It is a long chapter, but fully worth the effort.
But the issue is also nicely captured by your own analysis. As you show, any possible linear combination of the two hypotheses can be characterized by a single parameter, which is itself the probability that the next ball will be white. But when you have three hypotheses, you have two degrees of freedom. A single probability number no longer captures all there is to be said about what you know.
Popperians claim that we don’t need any theory of uncertainty to explain how
knowledge grows: uncertainty is irrelevant. That is an interesting claim don’t
you think? And if you care about the future of humanity, it is a claim that you
should take seriously and try to understand.
Popper’s views are out of date. I am somewhat curious about why anyone with access to the relevant information would fail to update their views—but that phenomenon is not that interesting. People fail to update all the time for a bunch of sociological reasons.
if I accept a bet from someone that a universal theory is true, would I ever have to pay out?
Check with the terms of the bet. Or...
Consider bets on when a bridge will fail. It might never fail—and if so, good for the bridge. However, if traders think it has a 50% chance of surviving to the end of the year, that tells you something. The market value of the bet gives us useful information about the expected lifespan of the bridge. It is just the same with scientific theories.
I claim that the distinction you make between events and theories is not nearly so clear-cut as you seem to think. You have already made the point that distinguishing between two or more apparent theories can readily be replaced by a parameterized theory. You restrict yourself to to the case where the parameterization is due to an “event”. I think most such cases can be tortured into such a view, particularly with your multiverse model. One of the earliest uses of probability theory was Laplace’s use in estimating orbital parameters for Jupiter and Saturn. If you take these parameters as themselves the theory, you would view it as illegitimate. If they are more akin to events, this seems fine. But your conception of events as “realizable” differently in the multiverse (i.e. all probabilities should be seen as indicial uncertainty) seems to be greatly underspecified. Given your example of GR as a theory rather than an event, why don’t you want to accept a multiverse model where GR really could hold in some universes, but not others? And of course, there’s a foundational issue that whatever multiverse model you take for events is itself a theory.
By multiverse I mean the everyday Everett/Deutsch one. I agree that the argument is a meta-theory about events and theories and that that meta-theory, like any theory, could have flaws.
Elliot has informed me that he doesn’t think he said: “humans can function as a Turing Machine by laboriously manipulating symbols”, except possibly in reply to a very specific question like “Give a short proof that humans have computational universality”.
Why do you say “people like Ellliot”? Elliot has his own views on things and shouldn’t be conflated with people who you think are like him. It seems to me you don’t understand his ideas so wouldn’t know what the people who are like him are like.
Human beings are universal knowledge creators: they can create any knowledge that any other knowledge creator can create.
For interesting definitions of ‘can’, perhaps. I know some humans who can’t create much of anything.
The only known tenable way of creating knowledge is by conjectures and refutations.
I’m not sure that counts as a ‘way of creating knowledge’. ‘Conjectures’ sounds to me like a black box which would itself contain the relevant bit.
Induction is a myth.
I’d want to know what you mean by ‘myth’. It’s worked so far, though that only counts as evidence for those of us blinded by the veil of Maya.
Theories are either true or false: there is no such thing as the probability that a theory is true.
Confirmation does not make a theory more likely or better supported—the only role of confirmation is to provide a ready stock of criticisms of rival theories.
Probability is in the mind. Theories are either true or false, and there is such a thing as the probability that a theory is true.
The most important knowledge is explanations.
I’m not sure what you mean by that.
There is no route to certain knowledge: we are all fallible.
This shows the remarks about ‘probability’ above to be merely a definitional dispute. Probability describes uncertainty, and you admit that we have uncertain knowledge.
We don’t need certain knowledge to progress: tentative, fallible, knowledge is just fine.
Human beings are universal knowledge creators: they can create any knowledge
that any other knowledge creator can create.
For interesting definitions of ‘can’, perhaps. I know some humans who can’t create
much of anything.
All human beings create knowledge—masses of it. Certain ideas can and do impair a person’s creativity, but it is always possible to learn and to change one’s ideas.
The only known tenable way of creating knowledge is by conjectures and refutations.
I’m not sure that counts as a ‘way of creating knowledge’. ‘Conjectures’ sounds to me
like a black box which would itself contain the relevant bit.
It’s not just conjectures, it’s “conjectures and refutations”. Knowledge is created by advancing conjectural explanations to solve a problem and then criticizing those conjectures in an attempt to refute them. The goal is to find a conjecture that can withstand all criticisms we can think of and to refute all rival conjectures.
Induction is a myth.
I’d want to know what you mean by ‘myth’. It’s worked so far, though that only counts
as evidence for those of us blinded by the veil of Maya.
No, it never worked. Not a bit. That’s what I mean by myth.
Theories are either true or false: there is no such thing as the probability that a
theory is true.
Confirmation does not make a theory more likely or better supported—the only role
of confirmation is to provide a ready stock of criticisms of rival theories.
Probability is in the mind. Theories are either true or false, and there is such a thing
as the probability that a theory is true.
Theories are objective. Whether you think a theory is true or false has no bearing on whether it is in fact true or false. Moreover, how do you assign a probability to a complex real-world theory like, say, multiversal quantum mechanics? What counts is whether the theory has stood up to criticism as an explanation to a problem or set of problems. If it has, who cares about how probable you think it is? It’s not the probability that you should care about, it’s the explanation.
The most important knowledge is explanations.
I’m not sure what you mean by that.
Above all else, we should try to find explanations for things; explanations are the most important kind of knowledge.
There is no route to certain knowledge: we are all fallible.
This shows the remarks about ‘probability’ above to be merely a definitional dispute.
Probability describes uncertainty, and you admit that we have uncertain knowledge.
Knowledge is always uncertain, yes, but it is impossible to objectively quantify the uncertainty. Put another way, you cannot know what you do not yet know. Theories can be wrong in all sorts of ways but you have no way of doing in advance how or if a theory will go wrong. It’s not a definitional dispute.
We don’t need certain knowledge to progress: tentative, fallible, knowledge is just
fine.
Ideas that should be taken more seriously by Less Wrong:
Human beings are universal knowledge creators: they can create any knowledge that any other knowledge creator can create.
The only known tenable way of creating knowledge is by conjectures and refutations.
Induction is a myth.
Theories are either true or false: there is no such thing as the probability that a theory is true.
Confirmation does not make a theory more likely or better supported—the only role of confirmation is to provide a ready stock of criticisms of rival theories.
The most important knowledge is explanations.
There is no route to certain knowledge: we are all fallible.
We don’t need certain knowledge to progress: tentative, fallible, knowledge is just fine.
Gee, I wonder what philosopher of science you have been reading. :)
I would suggest that you read through the sequences with an open mind—particularly on your point #4. If you find it impossible to open your mind on that point, then open it to the possibility that the word “probability” can have two different meanings and that your point #4 only applies to one of them. If you find it impossible to open your mind to the possibility that a word might have an alternative meaning which you have not yet learned, then please go elsewhere.
Regarding Popper, it is not so much that he is wrong, as that he is obsolete. We think we have learned that set of lessons and moved on to the next set of problems.
If you have already begun reading the sequences, and were motivated to give us this dose of Popper because Eliezer’s naive realism got on your nerves, well … All I can say is that it got on my nerves too, but if you keep reading you will find that EY is not nearly as epistemologically naive as it might seem in the early sequence postings.
No Popper is not obsolete and clearly the lessons of Popper have not been learnt by many: consider the people who have not yet understood that induction is a myth. Consider, also, the people who constantly misrepresent what Popper said like saying his philosophy is falsificationism or that he was a positivist or that he snuck induction in via the back door (you can find examples of these kind of mistakes discussed here). Popper’s ideas are in fact difficult for most people—they blow away the whole justificationist meta-context, a meta-context that permeates most people’s thinking. Understanding Popper requires that you take him seriously. David Deutsch did that and expanded on Popper’s ideas in a number of ways (you may be interested in a new book he has coming out called “The Beginning of Infinity”). He is another philosopher I follow closely. As is Elliot Temple (www.curi.us).
Thanks for the links and references. I will look into them. I urge you once more to work your way through the sequences. It appears you have something to teach us, but I doubt that you will be very successful until you have learned the local jargon, and become sufficiently familiar with our favorite examples to use them against us.
However, I have to say that I was a bit disconcerted by this:
Now if you told me that the standard definition of induction misrepresents the evidence-collection process, or that you know how to dissolve the problem of induction, well then I would be all ears. But when you say that “induction is a myth” I hear that as saying that everyone who has thought seriously on the topic, from Hume to the present, …, well, you seem to be saying that all those smart people were as deluded as the medieval philosophers who worried about angels dancing on the heads of pins.
See the thing is, I would have to keep having to upvote such arrogance and stupidity, just so the comment to which I am responding doesn’t disappear. And I don’t want to do that.
You do realize that Hume held that induction cannot be logically justified? He noticed there is a “problem of induction”. That problem was exploded by Karl Popper. Have you read what he has to say and taken seriously his ideas? Have you read and taken seriously the ideas of philosophers like David Deutsch, David Miller, and Bill Bartley? They all agree with Popper that:
Induction, i.e. inference based on many observations, is a myth. It is neither a psychological fact, nor a fact of ordinary life, nor one of scientific procedure - Karl Popper (Conjectures & Refutations, p 70).
Of course. That is why I mentioned him.
“Exploded”. My! What violent imagery. I usually prefer to see problems “dissolved”. Less metaphorical debris. And yes, I’ve read quite a bit of Popper, and admire much of it.
Nope, I haven’t.
You know, when giving page citations in printed texts, you should specify the edition. My 1965 Harper Torchbook paperback edition does not show Popper saying that on p 70. But, no matter.
One of the few things I dislike about Popper is that he doesn’t seem to understand statistical inference. I mean, he is totally clueless on the subject. It is not just that he isn’t a Bayesian—it seems he doesn’t “get” Pearson and Fisher either. Well, no philosopher gets everything right. But if he really thinks that “inference based on many observations” cannot happen—not just that it is frequently done wrong, but rather that it is impossible—then all I can say is that this is not one of Sir Karl’s better moments.
And if what he means is simply that we cannot infer absolute general truths from repeated observations, then I have to call him a liar for suggesting that anyone else ever suggested that we could make such inferences.
But, since you have been recommending philosophers to me, let me recommend some to you. I. J. Good is fun. Richard Jeffrey is not bad either. E.T. Jaynes explains quite clearly how one makes inferences based on observations—one observation or many observations. You really ought to look at Jaynes before coming to this forum to lecture on epistemology.
Perhaps you should know I have published papers where I have used Bayes extensively. I am well familiar with the topic (edit: though this doesn’t make me any kind of infallible authority). I was once enthusiastic about Bayesian epistemology myself. I now see it as sterile. Popperian epistemology—especially as extended by David Deutsch—is where I see fertile ground.
Cool. But more to the point, have you published, or simply written, any papers in which you explain why you now see it as sterile? Or would you care to recommend something by Deutsch which reveals the problems with Bayesianism. Something that actually takes notice of our ideology and tries to refute it will be received here much more favorably than mere diffuse enthusiasm for Popper.
The quote is from 3rd ed. 1968. You say you have read Popper, then you should not be surprised by the quote. Your response above is just the argument from incredulity. Do you have a better criticism?
I’m not surprised by the quote. I just couldn’t find it. It apparently wasn’t in 2nd edition. But my 2nd edition index had many entries for “induction, myth of _” so I don’t doubt you at all that Popper actually said it.
I am incredulous because I know how to do inference based on a single observation, as well as inference based on many. And so does just about everyone who posts regularly at this site. It is called Bayesian inference, and is not really all that difficult. Even you could do it, if you were to simply set aside your prejudice that
I have already provided references. You can find thousands more by Googling.
OK, tell me how you know in advance of having any theory what to observe?
BTW, please don’t assume things about me like asserting I hold prejudices. The philosophical position I come from is a full blown one. - it is no mere prejudice. Also, I’m quite willing to change my ideas if they are shown to be wrong.
Ok, I won’t assume that you believe, with Popper whom you quote, that inference based on many observations is impossible. I will instead assume that Popper is using the word “inference” very differently than it is used around here. And since you claim to be an ex-Bayesian, I will assume you know how the word is used here. Which makes your behavior up until now pretty inexplicable, but I will make no assumptions about the reasons for that.
Likewise, please do not assume that I believe that observation is neither theory-laden nor theory-directed. As it happens, I do not know in advance of a theory what to observe.
Of course, the natural thing for me to do now would be to challenge you to explain where theories come from in advance of observation. But why don’t we both just grow up?
If you have a cite for a careful piece of reasoning which will cause us to drop our Bayesian complaisancy and re-embrace Popper, please provide it and let us read the text in peace.
It sounds like Scurfield’s “cite for a careful piece of reasoning” are the works of Karl Popper, which you are also familiar with. I don’t have time to read the works of Karl Popper, but I have plenty of time to read blog comments about them. I’ve found every single comment in this thread interesting. Why discourage it?
I think the problem is a communication gap—”Bayesian” can mean different things to different people; and my best guess is that Scurfield converted from Laplace’s degree-of-belief approach to probability. Around here, though, the dominant Bayesian paradigm is Jaynes’, which takes the critiques of Bayes from the 1920 through the 1970s into account and digs through them to the epistemological bedrock below pretty well. Unless Scurfield has something new to say about Jaynes’ interpretation, his critiques aren’t that interesting to people who already know both Popper and Jaynes.
That can’t actually be everyone here. And I hope no one is offended if I say that Scurfield seems to “know Popper” to a greater degree than any of the other participants in this thread. Why the scorn for the guy and the conversation?
He certainly knows Popper better than me. I scorn the conversation because it is not stimulating me—not causing me to consider ideas I have never considered before. I scorn the guy (scorn may be a bit too strong here, but just go with it) because so far he has mostly presented slogans, rather than arguments. (Admitedly, I haven’t presented arguments either, but that is because his slogans strike me as either truisms or word games.)
The only thing I gained from this encounter was the link to the Critical Rationalism web site, where can be found links to writings by Popper and others. The CR site itself is, …, well, not great. For example, check out the “What is CR?” page where CR is contrasted with two other possible approaches to philosophy. Please actually check it out before continuing.
Now weren’t those subtle strawmen? :)
It occurs to me that one thing he could do which would be both interesting and useful would be to go through the sequences, adding comments critiquing Eliezer’s epistemology lessons from the viewpoint of Popper and/or CR. Who knows? I might frequently find myself agreeing with him.
Indeed, that’s why I am in favor of voting on old comments. Ideally, people can continue to leave criticisms on the sequences, and good ones will rise to the top over time.
Yes, I asked for clarification of the slogans and got more slogans, and asked for arguments supporting the claims and was given the claims again. I decided at that point to disengage.
Indeed—I hadn’t bothered to check out the site, but it seems to me that most of the discipline of Philosophy falls outside “CR”’s “three major schools”, and they’re pretending Popper invented philosophy. It’s really quite terrible.
If I may use another “slogan”: communication is difficult. And another: misunderstandings are common. When you asked for clarification I wasn’t sure what you wanted. I guessed and looks like I got it wrong. So you just withdraw? That’s very Un-Popperian.
Really? Care to give a quote?
It is a reasonable interpretation of the “three major schools” analysis down near the bottom of the “What is CR” page at the “Critical Rationalism” website. See if you can talk someone into cleaning up that bit of enthusiasm. As they say “It’s not helping”.
That’s a really high standard.
Hmmm. I never thought of that.
If you go as far as:
http://groups.yahoo.com/group/CriticalRationalism/
...you may see some names you recognise.
LOL. That made my day. Be sure to let me know if you run across TH anywhere.
Incidentally, have you looked in at sbe recently? Pretty sad.
I don’t see any people here that know both. Eliezer doesn’t appear to either. See here and here.
From the problem-situation. Theories arise out of problems.
And where do problems come from in advance of theories and obs...
Never mind. Someone else can carry on. I have other things to attend to.
A better phrasing for that might have been “certain knowledge is a myth.” What cannot be logically justified is reasoning from particular observations to certainty in universal truths. You’re commenting as if you are unaware of the positions and arguments linked from my previous reply, and perhaps Where Recursive Justification Hits Bottom . You have intelligent things to say, but you’re not going to be taken seriously here if you’re not aware of the pre-existing intelligent responses to them popular enough to amount to public knowledge.
No, that is not equivalent. Popper wrote that “inference based on many observations is a myth”. He is saying that we never reason from observations, never mind reasoning to certainty. In order to observe, you need theories. Without those, you cannot know what things you should observe or even make sense of any observation. Observation enables us to test theories, it never enables us to construct theories. Furthermore, Popper throws out the whole idea of justifying theories. We don’t need justification at all to progress. Judging from Where Recursive Justification Hits Bottom, this is something Eliezer has not fully taken on board (though I may be wrong). He sees the problem of the tu-quoque, but he still says [e]verything, without exception, needs justification. No, nothing can be justified. Knowledge advances not positively by justifying things but negatively by refuting things. Eliezer does see the importance of criticism, but my impression is that he doesn’t know Popper well enough.
For Yudkowsky on Popper, start here:
“Previously, the most popular philosophy of science was probably Karl Popper’s falsificationism—this is the old philosophy that the Bayesian revolution is currently dethroning.”
http://yudkowsky.net/rational/bayes
...and keep reading—at least as far as:
“On the other hand, Popper’s idea that there is only falsification and no such thing as confirmation turns out to be incorrect. Bayes’ Theorem shows that falsification is very strong evidence compared to confirmation, but falsification is still probabilistic in nature; it is not governed by fundamentally different rules from confirmation, as Popper argued.”
Yudhowsky gets a lot wrong even in a few sentences:
First, Popper’s philosophy cannot be accurately described as falsificationism—that is just a component of it, and not the most important component. Popperian philosophy consists of many inter-related ideas and arguments. Yudhowsky makes an error that Popperian newbies make. One suspects from this that Yudhowsky is making himself out to be more familiar with Popper than he actually is. His claim to be dethroning Popper would then be dishonest as he does not have detailed knowledge of the rival position. Also he is wrong that Popper is popular: he isn’t. Furthermore, Popper is familiar with Bayesian epistemology and actually discusses it in his books. So calling Popper’s philosophy old and making out that Bayesian epistemology is new is wrong also.
Popper never said theories can be definitely falsified. He was a thoroughgoing fallibilist and viewed falsifications as fallible conjectures. Also he said that theories can never be confirmed at all, not that they can be partially or probabilistically confirmed, which the above sentence suggests he said. Saying falsification is a special case of the Bayesian rules also doesn’t make sense: falsification is anti-induction whereas Bayesian epistemology is pro-induction.
Further comments on Yudhowski’s explanation of Bayes:
Science revolves around explanation and criticism. Most scientific ideas never get to the point of testing (which is a form of criticism), they are rejected via criticism alone. And they are rejected because they are bad explanations. Why is the emphasis in the quote solely on evidence? If science is a special case of Bayes, shouldn’t Bayes have something to say about explanation and criticism? Do you assign probabilities to criticism? That seems silly. Explanations and criticism enable us to understand things and to see why they might be true or false. Trying to reduce things to probabilities is to completely ignore the substance of explanations and criticisms. Instead of trying to get a probability that something is true, you should look for criticisms. You accept as tentatively true anything that is currently unproblematic and reject as tentatively false anything that is currently problematic. It’s a boolean decision: problematic or unproblematic.
Both bayesian induction (as we currently know it) and Popper fail my test for a complete epistemology.
The test is simple. Can I use the description of the formalism to program a real computer to do science? And it should, in theory, be able to bootstrap itself from no knowledge of science to our level.
If you were asked to bet on whether it was true or not, then you should assign a probability.
Scientists often do something like that when deciding how to allocate their research funds.
But then we have to develop a quantitative formalism for both beliefs and utilities. Is it really necessary to attack both problems at once?
Human beings don’t actually seem to have utility functions, all they really have are “preferences” i.e. a method for choosing between alternatives. But von Neumann and Morgenstern showed that under some conditions this is the same as having a utility function.
Now Scurfield is saying that human beings, even smart ones like scientists, don’t have prior probability distributions, all they really have is a database of claims and criticisms of those claims. Is there any result analogous to von Neumann-Morgenstern that says this is the same thing as having a prior, under conditions?
Yes. The question has been addressed repeatedly by a variety of people. John Maynard Keynes may have been the first. Notable formulations since his include de Finetti, Savage, and Jeffrey’s online book.
Discovering subjective probabilities is usually done in conjunction with discovering utilities by revealed preferences because much of the machinery (choices between alternatives, lotteries) is shared between the two problems. People like Jaynes who want a pure epistemology uncontaminated by crass utility considerations have to demand that their “test subjects” adhere to some fairly hard-to-justify consistency rules. But people like de Finetti don’t impose arbitrary consistency, instead they prove that inconsistent probability assignments lose money to clever gamblers who construct “Dutch books”.
I’d be interested in reading more about your views on this (unless you’re referring to Halpern’s papers on Cox’s theorem).
I’m not even familiar with Halpern’s work. The only serious criticism I have seen regarding the usual consistency rules for subjective probabilities dealt with the “sure thing rule”. I didn’t find it particularly convincing.
No, I have no trouble justifying a mathematical argument in favor of this kind of consistency. But not everyone else is all that convinced by mathematics. Their attention can be grabbed, however, by the danger of being taken to the cleaners by Dutch book professional bookies.
One of these days, I will get around to producing a posting on probability, developing it from what I call the “surprisal” of a proposition—the amount, on a scale from zero to positive infinity, by which you would be surprised upon learning that a proposition is true.
Prob(X) = 2^(-Surp(X)).
Surp(coin flip yields heads)= 1 bit.
Surp(A) + Surp(B|A) = Surp(A&B)
That last formula strikes me as particularly easy to justify (surprisals are additive). Given that and the first formula, you can easily derive Bayes law. The middle formula simply fixes the scale for surprisals. I suppose we also need a rule that Surp(True)=0
Actually “Surprisal” is a pretty standard term, I think.
Yudkowsky suggests calling it “absurdity” here
Cool! Saves me the trouble of writing that posting. :)
Absurdity is probably a better name for the concept. Except that it sounds objective, whereas amount of surprise more obviously depends on who is being surprised.
Wild. Is there an exposition of subjective expected utility better than wikipedia’s?
Jeffrey’s book, which I already linked, or any good text on Game theory. Myerson, for example, or Luce and Raiffa.
Agents can reasonably be expected to quantify both beliefs and utilities. How the ability to do that is developed—is up to the developer.
People are agents, and they are very bad at quantifying their beliefs and utilities.
I think that the contribution that Bayesian methodology makes toward good criticism of a scientific hypothesis is that to “do the math”, you need to be able to compute P(E|H). If H is a bad explanation, you will notice this when you try to determine (before you see E) how you would go about computing P(E|H). Alternately, you discover it when you try to imagine some E such that P(E|H) is different from P(E|not H).
No, you don’t assign probabilities to criticisms, as such. But I do think that every atomic criticism of a hypothesis H contains at its heart a conditional proposition of the form (E|H) or else a likelihood odds ratio P(E|H)/P(E|not H) together with a challenge, “So how would you go about calculating that?”
Incidentally, you also ought to look at some of the earlier postings where EY was, in effect, using naive Bayes classifiers to classify (i.e. create ontologies), rather than using Bayes’s theorem to evaluate hypotheses that predict. Also take a look at Pearl’s book to get a modern Bayesian view of what explanation is all about.
I like this point a lot. But it seems very convenient and sensible to say that some things are more problematic than others. And at least for certain kinds of claims it’s possible to quantify how problematic they are with numbers. This leads one (me at least) to want a formalism—for handling beliefs—that involves numbers, and Bayesianism is a good one.
What’s the conjectures-and-refutations way of handling claims like “it’s going to snow in February”? Do you think it’s meaningless or useless to attach a probability to that claim?
There is no problem with theories that make probabilistic predictions. But getting a probabilistic prediction is not tantamount to assigning a probability to the theory that made the prediction.
True. But you seem to be assuming that a “theory” has to be a universal law of nature. You are too attached to physics. But in other sciences, you can have a theory which is quite explanatory, but is not in any sense a “law”, but rather it is an event. Examples:
the theory that the moon was formed by a collision between the earth and a Mars-sized planetesimal.
the theory that modern man originated in Africa within the past 200,000 years and that the Homo erectus population outside of Africa did not contribute to our ancestry.
the theory that Napolean was poisoned with arsenic in St. Helena.
the “aquatic ape theory”
the endosymbiotic theory of the origin of mitochondria
the theory that the Chinese discovered America in 1421.
Probabilities can be assigned to these theories.
And even for universal theories, you can talk about the relative odds of competing theories being correct—say between a supersymetric GUT based on E6 and one based on E8. (Notice, I said “talk about the odds”, not “calculate them”) And you can definitely calculate how much one particular experimental result shifts those odds.
As you pointed out earlier, we have two ostensibly different ways of investigating the theory that the Chinese discovered America in 1421: the Popperian way, in which this theory and alternatives to it are criticized. And the Bayesian way, in which those criticisms are broken down into atomic criticisms, and likelihood ratios are attached and multiplied.
I’ve seen plenty of rigorous Popperian discussions but not very many very rigorous—or even modestly rigorous—Bayesian discussions, even on this website. One piece of evidence for the China-discovered-America theory is some business about old Chinese maps. How does a Bayesian go about estimating the likelihood ratio P(China discovered America | old maps) / P(China discovered America | no old maps)?
I think you want to ask about P(maps|discover) / P(no maps|discover). Unless both wikipedia and my intuition are wrong.
Does catching you in this error relieve me of the responsibility of answering the question? I hope so. Because I would want to instead argue using something like P(maps|discover) vs P(maps|not discover). That doesn’t take you all the way to P(discover), but it does at least give you a way to assess the evidential weight of the map evidence.
Now P(Sewing-Machine is a phony) = ?
Here’s another personal example of Bayesianism in action. Do you have a sense of how much you updated by? P(Richard Dawkins praises Steven Pinker | EP is bunk)/ P(Richard Dawkins praises Steven Pinker | EP is not bunk) is .5? .999? Any idea?
P(“Sewing Machine” is a nym) = 1.0
P(Sewing Machine has been disingenuous) = 0.5 and rising
P(Dawkins praises Pinker|EP is not bunk) is ill defined because
P(EP is not bunk) = ~0
but I have updated P(Dawkins believes EP is not bunk) to at least 0.5
I don’t know what “disingenuous” means.
The reason we would accept this theory as true is because it has survived criticism as an explanation while its rivals have not. If another rival theory is still in contention by also having survived criticism then there is a problem and this problem is not going to be resolved by computing, somehow, probabilities that the theories are true. To solve the problem you are going to have to come up with better criticisms or, possibly, alternative theories.
One difference between theories and events is that the counterfactuals of an event exist (in the multiverse). So it makes sense to talk about the probability of an event: the counterfactual events are real and occur to a greater or lesser measure in the multiverse. A false theory, by contrast, is simply false, it has no connection to reality. How do you assign anything other than an arbitrary probability to something that simply cannot be? Fortunately we don’t have to: we have Popperian epistemology.
Intrade knows that one! At the Higgs boson bet over 200 people think they know how to assign a probability to the issue.
http://www.intrade.com/jsp/intrade/common/c_cd.jsp?conDetailID=622297&z=1224442713385
The bet is for an event:
Shrug.
1: By using the counterfactuals in the Tegmark Level IV multiverse.
2: By giving it a probability of 0. If T is falsified, that means P(D|T)=0 - we obtained data that T claims is impossible. In this case, Bayes’ theorem sets P(T|D)=0. Bayesianism includes all correct thinking tools, including Popperian epistemology.
But is P(D|T) really 0? We could have made a mistake and not recorded the correct data. Certainly scientists in the past have done so, and thought that they falsified theories that they didn’t falsify. In this case, P(D|T) is very small but nonzero, and so is P(T|D) (unless p(D|~T) is also very small.)
3: You cannot avoid giving a probability. Because of Cox’s theorem, which says we must use probability theory to reason about uncertainty (although I must confess that the assumption that we must use a single real number to reason is rather strong.)
Counterfactuals of the sort of theory that Perplexed describe do exist in a Quantum Multiverse; why not assign them probabilities?
Anyway, why would you want to make your entire epistemology contingent on a particular theory of physics? It sounds like the CR would collapse if Copenhagen turned out to be correct.
Yes, counterfactuals do exist in some of Perplexed examples. There are alternative universes where Earth does not have a moon, or it does have a moon but it was not formed by a collision with a planetesimal. However, in this universe, the one we are observing now, the moon either was or it was not formed in such a way. There is no middle ground. Finding evidence consistent with the theory does not make the theory truer or more likely—what the evidence does is supply us with criticisms of rival theories.
CR is not independent from physics, nor can it be. The laws of physics entail the existence of universal computers and of universal knowledge creators. As David Deutsch has shown, there are deep connections between multiversal quantum physics and the theory of information. If Cophenhagen should turn out to be true it would impact on many things and not the least of which would be CR.
Events have probabilities. Theories don’t. Given some theory of meteorology, you can predict the probability it will snow in Feb. But you can’t say the probability that that theory of meteorology is true.
That’s true for frequentialist probabilities, but irrelevant in this context.
Did you read the context? Someone asked the Popperian view on giving a probability of future weather. So I answered that. What exactly do you think the context is?
The scientific method as a special case of Bayes theorem and whether the not directly experimental aspects can be mapped on some part of Bayesian reasoning. Now that you pointed it out I can see that you were only referring to the narrow sub-point in the great-gandparent and not the wider context, but it looked to me like you were also arguing that since theories don’t have (frequentlialist) probabilities Popperian reasoning about them couldn’t map to probabilistically framed Bayesian reasoning. Looking at the votes it seems I wasn’t alone in that (mis-)reading.
More from Yudkowsky on the philosophy of science:
http://lesswrong.com/lw/ig/i_defy_the_data/
The chance of a criticism being correct can unproblematically be assigned a probability.
A criticism can have many components, some of which are correct and some of which are incorrect. Breaking a criticism down into its components can be difficult/problematic.
Edit: The way I put that sounds stupid. Let me try again: occasionally, a pair of math papers are released, one purports to prove a conjecture, and one purports to disprove it. The authors then criticize each others papers (let’s say). Would you really characterize the task of assigning probabilities in this situation as “unproblematic”?
The point is that—if you were asked to bet on the criticism being correct—you would come up with some odds ratio.
Maybe you would do that. I would instead bog down in a discussion of whether the criticism was a nitpick or a “real” criticism. But I would be interested to see what odds ratio you come up with for this criticism being correct.
Heh—is that your criticism? - or did you get it from Douglas Hofstadter? ;-)
And in the math papers example, how exactly are you going to do that? Presumably you are going to go through the papers and the criticisms in detail and evaluate the content. And when you do that you are going to think of reasons why one is right and the other wrong. And then probabilities become irrelevent. It’s your understanding of the content that will enable you to choose.
Right—but you don’t “choose” - you assign probabilities. Rejecting something completely would be bad—because of:
http://lesswrong.com/lw/mp/0_and_1_are_not_probabilities/
I don’t think anyone is falling into this trap. It sounds like the Popperian version is replacing “true” and “false” by “tentatively true” and “tentatively false.”
“Tentatively true” and “tentatively false” sound a lot like probabilities which are not expressed in a format which is compatible with Bayes rule.
It is hard to see how that adds anything—but rather easy to see how it subtracts the ability to quantitatively analyse problems.
That’s what I said.
Edit: That refers to the first sentence only.
Theories are either true or false. The word “tentative” is there as an expression of fallibility. We cannot know if a theory is in fact true: it may contain problems that we do not yet know about. All knowledge is tentative. The word is not intended as a synonym for probability or to convey anything about probabilities.
Observers can put probabilities on the truth of theories. They can do it—and will do it—if you ask them to set odds and prepare to receive bets. Quantifying uncertainty allows it to be measured and processed.
It is true that knowledge is fallible—but some knowledge is more fallible than others—and if you can’t measure degrees of uncertainty, you will never develop a quantitative treatment of the subject. Philosophers of science realised this long ago—and developed a useful framework for quantifying uncertainty.
Scurfield missed his chance here. He should have asked when it becomes the case that those bets must be paid off, and offered the services of a Popper adept to make that kind of decision. Of course, the Popperite doesn’t rule that one theory is true, he rules that the other theory is refuted.
Short time limits don’t mean that agents can’t meaningfully assign probabilities to the truth of scientific theories—they just decrease the chances of the theories being proven wrong within the time limit a bit.
What is a time limit? Do actual bets on this sort of thing in Britain stipulate a time limit? As a Yank, I have no idea how betting ‘markets’ like this like this actually work.
Prediction markets/betting markets like Intrade or Betfair pretty universally set time limits on their bets. (Browse through Intrade sometime.) This does sometimes require changing the bet/prediction though—from ‘the Higgs boson will be found’ to ‘the Higgs boson will be found by 2020’. Not that this is a bad thing, mind you.
Do you have an answer to that point-that-should-have-been?
Not really. To the extent that we limit attention to theories of the form:
we Bayesians can never “cash in” on a bet that the theory is true—at least not using empirical evidence. All we can do is to continue trying to falsify the theory by experiments at more times, at more places, and for more values of x. As Popper prescribes. Our probabilities that the theory is true grow higher and higher, but they grow more and more slowly, and they can never reach unity.
However, both Bayesians and Popper fans can become pretty certain that such a theory is false—even without checking everywhere, everywhen, and forall x. Popper does not have a monopoly on refutations. Or conjectures either, for that matter.
There are no degrees of fallibility. We are simply fallible: that’s it. You have no way of knowing if a currently unproblematic theory is wrong, no matter how obvious the theory may seem, it may end up being spectacularly wrong.
I agree that we can quantity the uncertainty of events and that this is useful.
But theories are a different kettle of fish. Popperian epistemology tells us that we don’t need to know anything about the uncertainty of theories for knowledge to grow. Hence, one does not need to quantify the uncertainty of theories in order to write a knowledge creating computer program.
The thing is, we have a beautiful theory of uncertainty that deals with uncertainty over events, uncertainty about hypotheses—uncertainty about all beliefs, in fact—and it works just great.
Sure, we could go back to the dark days before Bayes, and struggle on with a boolean conception of certainty, and it probably wouldn’t be so bad that it would actually prevent knowledge from accumulating...
...but what possible reason would motivate us to take such a retrograde step?
I mean: how do you model equiprobable competing theories in such a framework.
How do you model induction? How do you model confirmation?
The answer seems to be that you don’t—you just deny the very existence of these phenomena!
I hope you can see how that is not a step forwards, from our point of view.
The thing is, we have a beautiful theory of uncertainty that deals with uncertainty over events, uncertainty about hypotheses—uncertainty about all beliefs, in fact—and it works just great.
Really? No exaggeration?
Popperian epistemology was created in the 20th century, after Bayes, so what do you mean by dark days. Certainty is not the lynchpin of knowledge creation and in fact has nothing to do with it. You completely devalue the role of explanations and criticisms. Who cares about probability when you have a good explanation that has withstood criticism?
Popperian philosophy is about problem-solving, explanation and criticism and these things are not in any way retrograde. Bayesian epistemology is rooted in the old philosophy of justificationism, a philosophy that Popperian epistemology overturned.
What you do is attempt further criticism of these theories and if that doesn’t progress come up with a meta-theory of what to do giving that you have two good candidate theories. It is always possible to come up with such a meta-theory.
Popper and others have given explanations—they don’t just deny without reason. Are you familiar with their arguments? There are real problems with the very concepts of induction and confirmation, problems that you seem not to have appreciated.
Popper showed that knowledge grows perfectly fine without concepts of uncertainty, induction, and confirmation. Yes, it is counter-intuitive—that is why most people do not get Popper.
Maybe you should present them, rather than playing ‘I can assert my philosopher is great more than you can’.
Bayesian scientific methods becoming more mainstream is a relatively modern phenomenon. See:
“Do we need to change the definition of science?”
http://postbiota.org/pipermail/tt/2008-May/002997.html
...for a recent overview.
The author of that link thinks Popper is falsification. I have already explained why that view of Popper is wrong. Have you been paying attention? And does the author of the link realize that the leading exponent of the multiverse is David Deutsch, who also happens to be the best living Popperian?
“Popper’s great and tireless efforts to expunge the word induction from scientific and philosophical discourse has utterly failed. Except for a small but noisy group of British Popperians, induction is just too firmly embedded in the way philosophers of science and even ordinary people talk and think.
Confirming instances underlie our beliefs that the Sun will rise tomorrow, that dropped objects will fall, that water will freeze and boil, and a million other events. It is hard to think of another philosophical battle so decisively lost.”—M.Gardner.
http://www.stephenjaygould.org/ctrl/gardner_popper.html
Gardner can be taken as seriously on Popper as he can on the MWI, i.e, not at all.
BTW, the sun does not rise in Murmansk in the middle of winter, live flies that are dropped do not fall, and water can be prevented from freezing in my car radiator by adding anti-freeze.
Edit: Here are some quotes from Gardner:
http://www.csicop.org/si/show/mind_at_play_an_interview_with_martin_gardner/
Carnap had a major influence on me. He persuaded me that all metaphysical questions are “meaningless” in the sense that they cannot be answered empirically or by reason. They can be defended only on emotive grounds. Carnap was an atheist, but I managed to retain my youthful theism in the form of what is called “fideism.” I like to call it “theological positivism,” a play on Carnap’s “logical positivism.
http://www.csicop.org/si/show/multiverses_and_blackberries/
As far as we can tell, universes are not as plentiful as even two blackberries. Surely the conjecture that there is just one universe and its Creator is infinitely simpler and easier to believe than that there are countless billions upon billions of worlds, constantly increasing in number and created by nobody. I can only marvel at the low state to which today’s philosophy of science has fallen
So why the downvote? Gardner doesn’t understand quantum physics and he doesn’t understand epistemology.
I still recommend Subjectively Objective, but I’m no longer confident that your inferential distance from the coverage there is small enough. Perplexed’s recommendation to read all the way through the sequences, or—even better—ET Jaynes’ Probability Theory: The Logic of Science—may be necessary. As he’s said, Critical Rationalism was an important step in the philosophy of science—but the field has moved beyond that to a rigorous, mathematically precise model of the amount of belief any rational agent must hold given identical priors and the same evidence—Popper’s Vs(a)=CT(a)-CTf(a) is not quantitative in this way.
That wasn’t intended to convince you; if you truly wish to subject your conjecture to criticism a contemplative reading of Jaynes is necessary. If you do happen to find Jaynes convincing, all is not lost—we still like Tarski here.
Popper obviously hadn’t read Wikipedia:
http://en.wikipedia.org/wiki/Inductive_reasoning
In what sense do you mean this exactly, and what evidence for it do you have? I’ve spoken to people like Elliot, but all they said was things like ‘humans can function as a Turing Machine by laboureously manipulating symbols’. Which is nice, but not really relevant to anything in real-time.
On a more general note, you should probably try to be a little clearer: ‘conjectures and refutations’ doesn’t really pick out any particular strategy from strategy-space, and neither does the phrase ‘explanation’ pick out anything in particular. Additionally, ‘induction’ is sufficiently different from what people normally think of as myths that it could do with some elaboration.
Similarly, some of these issues we do take seriously; we know we’re fallible, and it sounds like you don’t know what we mean by probability.
Finally, welcome to Less Wrong!
Edit: People, don’t downvote the parent; there’s no reason to scare the newbies.
Where ‘real-time’ can be taken literally to refer to time that is expected to exist in physics models of the universe.
Another way of saying it is that human beings can solve any problem that can be solved. Does that help?
Careful here—as I mentioned above, evidence never supports a theory, it just provides a ready stock of criticisms of rival theories. Let me give you an argument: If you hold that human beings are not universal knowledge creators, then you are saying that human knowledge creation processes are limited in some way, that there is some knowledge we cannot create. You are saying that humans can create a whole bunch of knowledge but whole realms of other knowledge are off limits to us. How does that work? Knowledge enables us to expand our abilities and that in turn enables us to create new knowledge and so on. Whatever this knowledge we can’t create is, it would have to be walled off from all this other expanding knowledge in a rather special way. How do you build a knowledge creation machine that only has the capability to create some knowledge? That would seem much much more difficult than creating a fully universal machine.
I don’t know what point Elliot was answering here, but I guess he is saying that humans are universal Turing Machines and illustrating that. He is saying that humans are universal in the sense that they can compute anything that can be computed. That is a different notion of universality to the one under discussion here (though there is a connection between the two types of universality). Elliot agrees that humans are universal knowledge creators and has written a lot about it (see, for example, his posts on The Fabric of Reality list).
‘Conjectures and refutations’ is an evolutionary process. The general methodology (or strategy, if you prefer) is: When faced with a problem try to come up with conjectural explanations to solve the problem and then criticise them until you find one (and only one) that cannot be knocked down by any known criticism. Take that as your tentative solution. I guess what you are looking for is an explanation of how human conjecture engines work? That is an unsolved problem. We do know some things, eg: no induction is involved.
Explanations are valuable: they help you understand something. Are you looking for an explanation of how we generate “explanations”? Again, unsolved problem.
It’s not really different. It’s something that people believe is true that in fact isn’t. Hume was the first to realize that there was a “problem of induction” and philosophers have for years and years been trying to justify induction. It took Karl Popper to realize that induction isn’t actually how we create knowledge at all: induction is a myth.
Yes, you are called “Less Wrong” after all! I was off-beam with that.
Actually, I am quite familiar with the Bayesian conception of probability. I just don’t think probability has a role in the realm of epistemology. Evidence does not make a theory more probable, not even from a subjective point of view. What evidence does, as I have said, is provide a stock of criticisms against rival theories. Also, evidence only goes so far: what really matters is how theories stand up to criticism as explanations. Evidence plays a role in that. I am quite happy to talk about the probability of events in the world, but events are different from explanatory theories. Apples and oranges.
What about the problem of building pyramids on alpha-centuri by 2012? We can’t, but aliens living there could.
More pressingly though, I don’t see why this is important. Have we been basing our arguments on an assumption that there are problems we can’t solve? Is there any evidence we can solve all problems without access to arbitrarily large amounts of computational power? Something like AIXI can solve pretty much anything, but not relevantly.
How about a neural network that can’t learn XOR?
The manner in which explanations are knocked down seems under-specified, if you’re not doing Bayesian updating.
Nope, I just don’t know what in particular you mean by ‘explanation’. I know what the word means in general, but not your specific conception.
Well, that’s different from there being no such thing as a probability that a theory is true: your initial assertion implied that the concept wasn’t well defined, whereas now you just mean it’s irrelevant. Either way, you should probably produce some actual arguments against Jaynes’s conception of probability.
Meta: You want to reply directly to a post, not its descendants, or the other person won’t get a notification. I only saw your post via the Recent Posts list.
Also, it’s no good telling people that they can’t use evidence to support their position because it contradicts your theory when the other people haven’t been convinced of your theory.
Criticism enables us to see flaws in explanations. What is under-specified about finding a flaw?
In your way, you need to come up with criticisms and also with probabilities associated with those criticisms. Criticisms of real world theories can be involved and complex. Isn’t it enough to expose a flaw in an explanatory theory? Must one also go to the trouble of calculating probabilities—a task that is surely fraught with difficulty for any realistic idea of criticism? You’re adding a huge amount of auxilliary theory and your evaluation is then also dependent on the truth of all this auxilliary theory.
My conception is the same as the general one.
You don’t seem to be actually saying very much then; is LW really short on explanations, in the conventional sense? Explanation seems well evidenced by the last couple of top level posts. Similarly, do we really fail to criticise one another? A large number of the comments seem to be criticisms. If you’re essentially criticising us for not having learn rationality 101 - the sort of rationality you learn as a child of 12, arguing against god—then obviously it would be a problem if we didn’t bare in mind the stuff. But without providing evidence that we succumb to these faults, it’s hard to see what the problem is.
Your other points, however, are substantive. If humans could solve any problem, or it was impossible to design an agent which could learn some but not all things, or confirmation didn’t increase subjective plausibility, these would be important claims.
Of course evidence makes theories more probable:
Imagine you have two large opaque bags full of beans, one 50% black beans and 50% white beans and the other full of white beans. The bags are well shaken, you are given one bag at random. You take out 20 beans—and they are all white.
That is clearly evidence that confirms the hypothesis that you have the bag full of white beans. If you had the “mixed” bag, that would only happen one time in a million.
Notice that the counterfactual event is possible (that you have the mixed bag). And even if you hold the bag full of white beans, the counterfactual event that you hold the mixed beans does occur elsewhere in the multiverse. This is what distinguishes events from theories. A false theory never obtains anywhere: it is simply false. So a theory being true or false is not at all like the situation with counterfactual events. You cannot assign anything objective to a false theory.
The actual theory you hold in your example is approximately the following: I have made a random selection from a bag and I know that I have been given one of two bags: one 50% black beans and 50% white beans and the other full of white beans and: I have been honestly informed about the setup, am not being tricked, no mistakes have been made etc. This theory predicts that if I take 20 white beans out of the bag, then the chance of that would be one in a million if I had the mixed bag. Do you understand? The real situation is that you have a theory that is making probabilistic predictions about events and, as I have said several times, I have no problem with probabilistic predictions of theories about events.
As briefly as possible:
Firstly, this seems like a step forwards to me. You seem to agree that induction and confirmation are fine 90% of the time. You seem to agree that these ideas work in practice—and are useful—including in some realms of knowledge—such as knowledge relating to which bag is in front of you in the above example. This puts your anti-induction and anti-confirmation statements into a rather different light, IMO.
Confirmation theory has nothing to do with multiverses. It applies equally well for agents in single deterministic universes—such as can be modelled by cellular automata. So, reasoning that depends on the details of multiverse theories is broken from the outset. Imagine evidence for wavefunction collapse was found. Not terribly likely—but it could happen—and you don’t want your whole theory of epistemology to break if it does!
Treating uncertainty about theories and uncertainty about events differently is a philosophical mistake. There is absolutely no reason to do it—and it gets people into all kinds of muddles.
We have a beautiful theory of subjective uncertainty that applies equally well to uncertainty about any belief—whether it refers to events, or scientific theories. You can’t really tease these categories apart anyway—since many events are contingent upon the truth of scientific theories—e.g. Higgs boson observations. Events are how physical law is known to us.
Instead of using one theory for hypotheses about events and another for hypotheses about universal laws you should—according to Occam’s razor—be treating them in the same way—and be using the same underlying general theory that covers all uncertain knowledge—namely the laws of subjective probability.
“Bayesian Confirmation Theory”
http://plato.stanford.edu/entries/epistemology-bayesian/#BayTheBayConThe
Tim—In the example we have been discussing, no confirmation of the actual theory (the one I gave in approximate outline) happens. The actual theory makes probabilistic predictions about events (it also makes non-probabilistic predictions) and tells you how to bet. Getting 20 white beans doesn’t make the actual theory any more probable—the probability was a prediction of the theory. Note also that a theory that you are being tricked might recommend that you choose the mixed bag when you get 20 white beans. Lots of theories are consistent with the evidence. What you need to look for is things to refute the possible theories. If you are concerned with confirmation, then the con man wins.
So I am not agreeing that induction and confirmation are fine any percentage of the time (how did you get that 90% figure?). When you consider the actual possible theories of the example, all that is happening is that you have explanatory theories that make predictions, some probabilistic, and that tell you how to bet. The theories are not being induced from evidence and no confirmation takes place.
You haven’t explained how we assign objective probabilities to theories that are false in all worlds.
We don’t assign objective probabilities, full stop.
What you’re talking about here is a strategy for avoiding bias which Bayesians also use. It is not a fundamental feature of any particular epistemology.
I think you are too lost for me :-(
You don’t seem to address the idea that multiverse theories are an irrelevance—and that in a single deterministic automaton, things work just the same way.
Indeed, scientists don’t even know which (If any) laws of physics are true everywhere, and which depend on the world you are in.
You don’t seem to address the idea that we have a nice general theory that covers all kinds of uncertainty, and that no extra theory to deal with uncertainty about scientific hypotheses is needed.
If you don’t class hypotheses about events as being “theories”, then I think you need to look at:
http://en.wikipedia.org/wiki/Scientific_theory
Also, your challenge doesn’t seem to make much sense. The things people assign probabilities to are things they are uncertain about. If you tell me a theory is wrong, it gets assigned a low probability. The interesting cases are ones where we don’t yet know the answer—like the clay theory of the origin of life, the orbital inclination theory of glacial cycles—and so on.
Distinguishing between scientific theories and events in the way that you do apparently makes little sense. Events depend on scientific theories. Scientific theories predict events. Every test of a scientific theory is an event. Observing the perihelion precession of Mercury was an event. The observation of the deflection of light by the Sun during an eclipse was an event. If you have probabilities about events which are tests of scientific theories, then you automatically wind up with probabilities about the theories that depend on their outcome.
Basically agents have probabilities about all their beliefs. That is Bayes 101. If an agent claims not to have a probability about some belief, you can usually set up a bet which reveals what they actually think about the subject. Matters of fundamental physics are not different from “what type of beans are in a bag”—in that respect.
Yes, scientific theories predict events. So there is a distinction between events and theories right? If the event is observed to occur, all that happens is that rival theories that do not predict the event are refuted. The theory that predicted the event is not made truer (it already is either true or false). And there are always an infinite number of other theories that predict the same event. So observing the event doesn’t allow you to distinguish among those theories.
In the bean bag example you seem to think that the rival theories are “the bag I am holding is mixed” and “the bag I am holding is all white”. But what you actually have is a single theory that makes predictions about these two possible events. That theory says you have a one-in-a-million chance of holding the mixed bag.
No, General Relativity being true or false is not like holding a bag of white beans or holding a bag of mixed beans. The latter are events that can and do obtain: They happen. But GR is not true in some universes and false in others. It is either true or false. Everywhere. Furthermore, we accept GR not because it is judged most likely but because it is the best explanation we have.
Popperians claim that we don’t need any theory of uncertainty to explain how knowledge grows: uncertainty is irrelevant. That is an interesting claim don’t you think? And if you care about the future of humanity, it is a claim that you should take seriously and try to understand.
If you are still confused about my position, why don’t you try posting some questions on one of the following lists:
http://groups.yahoo.com/group/Fabric-of-Reality/
http://groups.yahoo.com/group/criticalrationalism/
It might be useful for other Popperians to explain the position—perhaps I am being unclear in some way.
Edit: Just because people might be willing to place bets is no argument that the epistemological point I am making is wrong. What makes those people infallible authorities on epistemology? Also, if I accept a bet from someone that a universal theory is true, would I ever have to pay out?
That’s a really powerful general argument against Bayesianism that I hadn’t considered before: any prior (edit: I should have said “prior information”) necessarily constitutes a hypothesis in which you have confidence 1.
I don’t think that statement makes sense; you seem to be mixing levels—the prior is a distribution over how the world could actually be, not over other distributions. It shouldn’t make sense to speak of your prior’s confidence in itself.
You have an explanatory theory that makes predictions about the events, but it is not the only possible explanatory theory. If someone offers to play the bean bag game with you on the street, then things might not be as they seem and your theory would be no good as an explanation of how to bet. Science is like that—what is actually going on might not be what you think, so you look for flaws and realize that one’s confidence is no guide to the truth.
If your confidence in your prior were 1, you would never be able to update it. But, it is true that if your prior distribution of probabilities over various hypotheses assigns 0 or 1 probability to a group of hypotheses, you will never be able to accrue enough evidence to change that. This is not a weakness of Bayesianism, because there is no other method of reasoning which will allow you to end up on a conclusion which you at no point considered as a possibility.
Did you read the quoted text? Inability to update is the whole point of my concern; but it in no way implies that my confidence in a particular outcome will never change.
Perhaps you’re confusing probabilities for priors. (edit: I was misusing my terms: I meant “prior probabilities” and “prior information” respectively.)
I think that the problem is that EY has introduced non-standard terminology here. Worse, he blames it on Jaynes, who makes no such mistake. I just looked it up.
There are two concepts here which must not be confused.
a priori information, aka prior information, aka background information
prior probabilities, aka priors (by everyone except EY. Jaynes dislikes this but acquiesces).
Prior information does indeed constitute a hypothesis in which you have complete confidence. I agree this is something of a weakness—a weakness which is recognized implicitly in such folklore as “Cromwell’s rule” Prior information cannot be updated.
Prior probabilities (frequently known simply as priors) can be updated. In a sense, being updated is their whole purpose in life.
This is exactly what’s going on. Thank you.
I apologize for my confused terminology.
You are welcome. Unfortunately, I was wrong. Or at least incomplete.
I misinterpreted what EY was saying in the posting you cited. He was not, as I mistakenly assumed, saying that prior probabilities should not be called priors. He was instead talking about a third kind of entity which should not be confused with either of the other two.
Prior distributions over hypotheses, which Eliezer wishes to call simply “priors”
But there is not a confusion with referring to both prior probabilities and prior distributions as simply priors because a prior probability is simply a special case of a prior distribution. A probability is simply a distribution over a set of two competing hypotheses—only one of which can be true.
Bayes theorem in its usual form applies only to simple prior probabilities. It tells you how to update the probability. In order to update a prior distribution, you effectively need to use Bayes’s theorem multiple times—once for each hypothesis in your set of hypotheses.
So what is that 1⁄2 number which Eliezer says is definitely not a prior? It is none of the above three things. It is something harder to describe. A statistic over a distribution. I am not even going to try to explain what that means. Sorry for any confusion I may have created. And thx to Sniffnoy and timtyler for calling my attention to my mistake.
I’m not convinced that there’s a meaningful difference between prior distributions and prior probabilities.
Going back to the beans problem, we have this:
This can easily be “flattened” into a single, more complex, probability distribution:
If we wish to consider multiple draws, we can again flatten the total event into a single distribution:
Translating the “what is that number” question into this situation, we can ask: what do we mean when we say that we are 5⁄8 sure that we will draw two white beans? I would say that it is a confidence; the “event” that has 5⁄8 probability is a partial event, a lossy description of the total event.
There isn’t when you have only two competing hypotheses. Add a third hypothesis and you really do have to work with distributions. Chapter 4 of Jaynes explains this wonderfully. It is a long chapter, but fully worth the effort.
But the issue is also nicely captured by your own analysis. As you show, any possible linear combination of the two hypotheses can be characterized by a single parameter, which is itself the probability that the next ball will be white. But when you have three hypotheses, you have two degrees of freedom. A single probability number no longer captures all there is to be said about what you know.
In retrospect, it’s obvious that “probability” should refer to a real scalar on the interval [0,1].
Everyone calls prior probabilities “priors”—including: http://yudkowsky.net/rational/bayes
Uh, what? No it doesn’t. If your confidence in your priors was that high, they would never shift.
Popper’s views are out of date. I am somewhat curious about why anyone with access to the relevant information would fail to update their views—but that phenomenon is not that interesting. People fail to update all the time for a bunch of sociological reasons.
Check with the terms of the bet. Or...
Consider bets on when a bridge will fail. It might never fail—and if so, good for the bridge. However, if traders think it has a 50% chance of surviving to the end of the year, that tells you something. The market value of the bet gives us useful information about the expected lifespan of the bridge. It is just the same with scientific theories.
I claim that the distinction you make between events and theories is not nearly so clear-cut as you seem to think. You have already made the point that distinguishing between two or more apparent theories can readily be replaced by a parameterized theory. You restrict yourself to to the case where the parameterization is due to an “event”. I think most such cases can be tortured into such a view, particularly with your multiverse model. One of the earliest uses of probability theory was Laplace’s use in estimating orbital parameters for Jupiter and Saturn. If you take these parameters as themselves the theory, you would view it as illegitimate. If they are more akin to events, this seems fine. But your conception of events as “realizable” differently in the multiverse (i.e. all probabilities should be seen as indicial uncertainty) seems to be greatly underspecified. Given your example of GR as a theory rather than an event, why don’t you want to accept a multiverse model where GR really could hold in some universes, but not others? And of course, there’s a foundational issue that whatever multiverse model you take for events is itself a theory.
By multiverse I mean the everyday Everett/Deutsch one. I agree that the argument is a meta-theory about events and theories and that that meta-theory, like any theory, could have flaws.
Elliot has informed me that he doesn’t think he said: “humans can function as a Turing Machine by laboriously manipulating symbols”, except possibly in reply to a very specific question like “Give a short proof that humans have computational universality”.
Why do you say “people like Ellliot”? Elliot has his own views on things and shouldn’t be conflated with people who you think are like him. It seems to me you don’t understand his ideas so wouldn’t know what the people who are like him are like.
For interesting definitions of ‘can’, perhaps. I know some humans who can’t create much of anything.
I’m not sure that counts as a ‘way of creating knowledge’. ‘Conjectures’ sounds to me like a black box which would itself contain the relevant bit.
I’d want to know what you mean by ‘myth’. It’s worked so far, though that only counts as evidence for those of us blinded by the veil of Maya.
Probability is in the mind. Theories are either true or false, and there is such a thing as the probability that a theory is true.
I’m not sure what you mean by that.
This shows the remarks about ‘probability’ above to be merely a definitional dispute. Probability describes uncertainty, and you admit that we have uncertain knowledge.
True that!
Welcome to Less Wrong
ETA: Reminder that we have a rough community norm against downvoting first posts when they seem to be in good faith.
All human beings create knowledge—masses of it. Certain ideas can and do impair a person’s creativity, but it is always possible to learn and to change one’s ideas.
It’s not just conjectures, it’s “conjectures and refutations”. Knowledge is created by advancing conjectural explanations to solve a problem and then criticizing those conjectures in an attempt to refute them. The goal is to find a conjecture that can withstand all criticisms we can think of and to refute all rival conjectures.
No, it never worked. Not a bit. That’s what I mean by myth.
Theories are objective. Whether you think a theory is true or false has no bearing on whether it is in fact true or false. Moreover, how do you assign a probability to a complex real-world theory like, say, multiversal quantum mechanics? What counts is whether the theory has stood up to criticism as an explanation to a problem or set of problems. If it has, who cares about how probable you think it is? It’s not the probability that you should care about, it’s the explanation.
Above all else, we should try to find explanations for things; explanations are the most important kind of knowledge.
Knowledge is always uncertain, yes, but it is impossible to objectively quantify the uncertainty. Put another way, you cannot know what you do not yet know. Theories can be wrong in all sorts of ways but you have no way of doing in advance how or if a theory will go wrong. It’s not a definitional dispute.
OK, we agree on that!
Probability is subjectively objective. All conjectures/models are wrong, but some are useful to the extent that they successfully constrain expected experience.