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.
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.