Manfred

Karma: 10,786

Manfred Jul 31, 2017, 10:24 PM
0 points
in reply to: Oscar_Cunningham’s comment on: Open thread, July 31 - August 6, 2017
Cool insight. We’ll just pretend constant density of 3M/4r^3.

This kind of integral shows up all the time in E and M, so I’ll give it a shot to keep in practice.

You simplify it by using the law of cosines, to turn the vector subtraction 1/|r-r’|^2 into 1/(|r|^2+|r’|^2+2|r||r’|cos(θ)). And this looks like you still have to worry about integrating two things, but actually you can just call r’ due north during the integral over r without loss of generality.

So now we need to integrate 1/(r^2+|r’|^2+2r|r’|cos(θ)) r^2 sin(θ) dr dφ dθ. First take your free 2π from φ. Cosine is the derivative of sine, so substitution makes it obvious that the θ integral gives you a log of cosine. So now we integrate 2πr (ln(r^2+|r’|^2+2r|r’|) - ln(r^2+|r’|^2-2r|r’|)) / 2|r’| dr from 0 to R. Which mathematica says is some nasty inverse-tangent-containing thing.

Okay, maybe I don’t actually want to do this integral that much :P

Manfred Jul 31, 2017, 6:39 PM
0 points
in reply to: entirelyuseless’s comment on: Sleeping Beauty Problem Can Be Explained by Perspective Disagreement (IV)
To spell it out:

Beauty knows limiting frequency (which, when known, is equal to the probability) of the coin flips that she sees right in front of her will be equal to one-half. That is, if you repeat the experiment many times (plus a little noise to determine coin flips), then you get equal numbers of the event “Beauty sees a fair coin flip and it lands Heads” and “Beauty sees a fair coin flip and it lands Tails.” Therefore Beauty assigns ⁵⁰⁄₅₀ odds to any coin flips she actually gets to see.

You can make an analogous argument from symmetry of information rather than limiting frequency, but it’s less accessible and I don’t expect people to think of it on their own. Basically, the only reason to assign thirder probabilities is if you’re treating states of the world given your information as the basic mutually-exclusive-and-exhaustive building block of probability assignment. And the states look like Mon+Heads, Mon+Tails, and Tues+Tails. If you eliminate one of the possibilities, then the remaining two are symmetrical.

If it seems paradoxical that, upon waking up, she thinks the Monday coin is more likely to have landed tails, just remember that half of the time that coin landed tails, it’s Tuesday and she never gets to see the Monday coin being flipped—as soon as she actually expects to see it flipped, that’s a new piece of information that causes her to update her probabilities.

Manfred Jul 30, 2017, 4:09 AM
0 points
in reply to: Xianda_GAO_duplicate0.5321505782395719’s comment on: Sleeping Beauty Problem Can Be Explained by Perspective Disagreement (IV)

According to SSA beauty should update credence of H to ²⁄₃ after learning it is Monday.

I always forget what the acronyms are. But the probability of H is ¹⁄₂ after learning it’s Monday, any any method that says otherwise is wrong, exactly by the argument that you can flip the coin on monday right in front of SB, and if she knows it’s Monday and thinks it’s not a ⁵⁰⁄₅₀ flip, her probability assignment is bad.

Manfred Jul 28, 2017, 10:19 PM
0 points
on: Sleeping Beauty Problem Can Be Explained by Perspective Disagreement (IV)

He proposes the coin toss could happen after the first awakening. Beauty’s answer ought to remain the same regardless the timing of the toss. A simple calculation tells us his credence of H must be ¹⁄₃. As SSA dictates this is also beauty’s answer. Now beauty is predicting a fair coin toss yet to happen would most likely land on T. This supernatural predicting power is a conclusive evidence against SSA.

So how do you get Beauty’s prediction? If at the end of the first day you ask for a prediction on the coin, but you don’t ask on the second day, then now Beauty knows that the coin flip is, as you say, yet to happen, and so she goes back to predicting ⁵⁰⁄₅₀. She only deviates from ⁵⁰⁄₅₀ when she thinks there’s some chance that the coin flip has already happened.

Sometimes people absolutely will come to different conclusions. And I think you’re part of the way there with the idea of letting people talk to see if they converge. But I think you’ll get the right answer even more often if you set up specific thought-experiment processes, and then had the imaginary people in those thought experiments bet against each other, and say the person (or group of people all with identical information) who made money on average (where “average” means over many re-runs of this specific thought experiment) had good probabilities, and the people who lost money had bad probabilities.

I don’t think this is what probabilities mean, or that it’s the most elegant way to find probabilities, but I think it’s a pretty solid and non-confusing way. And there’s a quite nice discussion article about it somewhere on this site that I can’t find, sadly.

Manfred Jul 28, 2017, 7:46 PM
0 points
in reply to: Xianda_GAO_duplicate0.5321505782395719’s comment on: Sleeping Beauty Problem Can Be Explained by Perspective Disagreement (II)
Sorry for the slow reply.

The 8 rooms are definitely the unbiased sample (of your rooms with one red room subtracted).

I think you are making two mistakes:

First, I think you’re too focused on the nice properties of an unbiased sample. You can take an unbiased sample all you want, but if we know information in addition to the sample, our best estimate might not be the average of the sample! Suppose we have two urns, urn A has 10 red balls and 10 blue balls, while urn B has 5 red balls and 15 blue balls. We choose an urn by rolling a die, such that we have a ⁵⁄₆ chance of choosing urn A and a ¹⁄₆ chance of choosing urn B. Then we take a fair, unbiased sample of 4 balls from whatever urn we chose. Suppose we draw out 1 red ball and 3 blue balls. Since this is an unbiased sample, does the process that you are calling “statistical analysis” have to estimate that we were drawing from urn B?

Second, you are trying too hard to make everything about the rooms. It’s like someone was doing the problem with two urns from the previous paragraph, but tried to mathematically arrive at the answer only as a function of the number of red balls drawn, without making any reference to the process that causes them to draw from urn A vs. urn B. And they come up with several different ideas about what the function could be, and they call those functions “the Two-Thirds-B-er method” and “the Four-Tenths-B-er method.” When really, both methods are incomplete because they fail to take into account what we know about how we picked the urn to draw from.

To answer the last part of your statement. If beauty randomly opens 8 doors and found them all red then she has a sample of pure red. By simple statistics she should give R=81 as the estimation. Halfer and thirders would both agree on that. If they do a bayesian analysis R=81 would also be the case with the highest probability. I’m not sure where 75 comes from I’m assuming by summing the multiples of probability and Rs in the bayesian analysis? But that value does not correspond to the estimation in statistics. Imagine you randomly draw 20 beans from a bag and they are all red, using statistics obviously you are not going to estimate the bag contains 90% red bean.

Think of it like this: if Beauty opens 8 doors and they’re all red, and then she goes to open a ninth door, how likely should she think it is to be red? 100%, or something smaller than 100%? For predictions, we use the average of a probability distribution, not just its highest point.

Manfred Jul 27, 2017, 12:12 AM
0 points
on: Type Theory quick question
The HoTT book is pretty readable, but I’m not in a position to evaluate its actual goodness.

Manfred Jul 25, 2017, 7:46 PM
0 points
on: Steelmanning as an alternative to Rationalist Taboo
In your example, I think Bob is doing something unrelated to rationalist Taboo.

In the actual factual game of Taboo, you replace a word with a description that is sufficient to tell your team what the original word is. In rationalist Taboo, you replace a word with a description that is sufficient to convey the ideas you were trying to convey with the original word.

So if Bob tries to taboo “surprise” as “the feeling of observing a low-probability event,” and Alice says “A license plate having the number any particular number is low probability—is it surprising?,” Bob should think “Oh, the description I replaced ‘surprise’ with did not convey the same thing as the word ‘surprise’. I need to try tabooing it differently.”

This works better when you’re trying to taboo the usage of a word in a specific context, because the full meaning of a word is very very complicated (though trying to make definitions can still be a fun and profitable game, I agree), but when you look at how you’ve used it in just one sentence, then you have some hope of pinning down what you mean by it to your satisfaction.

Manfred Jul 24, 2017, 6:31 PM
0 points
on: Is Altruism Selfish?
And yet, people, when giving examples of selfishness, don’t just sample the entirety of human behavior. They point out a specific sort of behavior. Or when naming optimization functions, they might call one function “greedy,” even though all functions tautologically do what they do. So clearly people have some additional criteria for everyday use of the word not captured by the extremely simple definition in this post.

Manfred Jul 24, 2017, 6:01 PM
3 points
in reply to: tukabel’s comment on: How long has civilisation been going?
First, I checked out the polling data on interracial marriage. Every 10 years the approval rating has gone up by ~15 percentage points. I couldn’t find a concise presentation of the age-segregated data from now vs. in the past, but 2007 and 1991 were available, and they look consistent with over 80% of the opinion change being due to old people dying off. This surprised me, I expected to see more evidence of people changing their mind.

Now look at gay marriage.. It’s gained at ~18 points per 10 years. This isn’t too different from 15, so maybe this is people dying off too. And indeed it seems to be mostly the case—except the in the last 10 years, where the gains don’t follow the right age pattern, indicating that of 18 points of gain, about 40% may actually involve people changing their minds.

Manfred Jul 21, 2017, 9:29 PM
4 points
0
on: Can anyone refute these arguments that we live on the interior of a hollow Earth?
“Refute” is usually not an objective thing—it’s a social thing. You can probably prove to yourself that pi=3 is false, but if you write “pi=3″ on a sheet of paper, no argument will make the ink rearrange itself to be correct.

This is one of the problems with a falsificationist idea of scientific progress, where we never prove theories true but make progress by proving them false. If evidence against a theory appears (e.g. the ability to see different stars from different parts of the earth might be thought of as “refuting” the idea of a flat earth), a proponent of that theory never has to give up on it. They can just patch the theory. Maybe light does a special little dance to make all the observations look like we’re looking out at a universe, etc. If you try to refute someone, they can just refuse to be refuted and add another patch to their theory.

After doing some reading, I feel like this guy actually does a pretty admirable job of seeing open questions and admitting ignorance. For example, he doesn’t know about the coriolis effect, so he calls it “a mysterious thing that happens to objects falling down mineshafts” and wonders whether it could cause an error in the readings of plumb bobs hanging down a mineshaft. Again, I think this is a good thing, though not as good as knowing about the coriolis effect before trying to understand the structure of the cosmos. The trouble seems mostly to be that he’s read a lot of books that are full of shit, and believes them.

Manfred Jul 20, 2017, 5:19 PM
0 points
on: Sleeping Beauty Problem Can Be Explained by Perspective Disagreement (II)
Are the rooms biased towards blue? Let’s do this with balls from an urn.

Suppose there are four balls in an urn. 1,2,3,or 4 of them can be red, I’ve chosen the number (R) by a die roll. The rest are blue. We draw the balls from the urn in order.

Conditioned on the first ball being red, do we see a similar effect between the first two and last two balls?

Conditioning on the first ball being red means that it’s more likely that my die roll was high—the distribution becomes proportional to R. Then if I see the second ball is blue, the probability distribution over R is proportional to R*(4-R), which is symmetrical about 2. So I expect that about one of the other balls will be red. Which means that the “other observed room” (ball 2) is systematically less red than an unknown room.

So this effect also occurs with drawing balls out of an urn! But what does it mean in this case? Does it mean we made a mistake with our math? No—you can do the experiment yourself easily. Hmm. Let’s go back to your three possible desiderata:

A: Are the first two balls an unbiased sample of the balls? No—we conditioned that the first one was red. Even if R=1, we cannot get two blues in the first two balls. That’s bias.

Note that when conditioning on the fact that the first ball was red, we used Bayes’ rule for conditioning. We didn’t assume that someone looked into the urn and pulled out a red ball, instead there was in some sense a fair process that “just happened” to give us a red ball first. But this does not mean that the first two balls are an unbiased sample. The fairness of the hypothetical process that gave us a red ball first does not carry over in any important way once we condition on it giving us a specific result. I think this may be a tricky point.

B: The first ball is guaranteed to be red. Yup.

C: The second ball is an unbiased sample of the other three balls. Yes. Even though it’s less red than unknown ball in this example. But on the other hand, if it was red, it would be more red than an unknown ball.

Wait. I think you’re not doing the math properly, which means that last statement doesn’t hold in your post.

Back to the rooms and Sleeping Beauty. I think you’re claiming that no matter how many red rooms S.B. sees, she expects even more, and therefore expects her surroundings to always be magically blue-biased. But if you do things correctly, then if Sleeping Beauty opens up 8 more rooms and they’re all red, she doesn’t jump straight to a belief that R=81. Instead, she thinks that R is on average somewhere around 75, and also she got a red-biased batch of rooms. But this only works if you keep track of these possibilities for R.

Manfred Jul 17, 2017, 8:05 PM
0 points
in reply to: Thomas’s comment on: Open thread, Jul. 17 - Jul. 23, 2017
Note that your force grows unboundedly in N, so close to zero you have things that are arbitrarily heavy compared to their distance. So what this paradox really is about, is alternating series’ that grow with N, and whether we can say that they add up to zero.

If we call the force between the first two bodies f12, then the series of internal forces on this system of bodies (using negative to denote vector component towards zero) looks like -f12+f12-f23+f23-f13+f13-f34..., where, again, each new term is bigger than the last.

If you split this sum up by interactions, it’s (-f12+f12)+(-f23+f23)+(-f13+f13)..., so “obviously” it adds up to zero. But if you split this sum up by bodies, each term is negative (and growing!) so the sum must be negative infinity.

The typical physicist solution is to say that open sets aren’t physical, and to get the best answer we should take the limit of compact sets.

Manfred Jul 6, 2017, 5:21 PM
0 points
in reply to: cousin_it’s comment on: Steelmanning the Chinese Room Argument
Hm, this is an interesting question to think about. I lean more towards the camp of construing consciousness as broad and pretty easy to attain, but only a small part of a mind’s value. As long as we can push down the probability of lookup tables and push up the probability of self-reflection and abstract thinking.

Weird example I’d label as conscious: an AI that can observe us, trying to fool us in a particular way: Our brains compute expectations of what kinds of things a conscious correspondent would say, then the AI can observe these expectations and compute something consistent both with our expectations and its past responses. Most of the computation of a mind is there, but packaged differently and spread over multiple media—the text responses no longer reflect consciousness if the AI loses its observation channel.

Manfred Jul 6, 2017, 2:57 PM
1 point
on: Steelmanning the Chinese Room Argument
What if the person claims to be able to add numbers? If you ask them about 2+2 and they answer 4, maybe they were pre-ordered with that response, but if you get them to add a few dozen poisson-distributed numbers, maybe you start believing they’re actually implementing the algorithm. This relies on the important distinction between telling two things apart certainly and gathering evidence.

Manfred Jun 29, 2017, 6:06 PM
2 points
in reply to: Erfeyah’s comment on: Open thread, June 26 - July 2, 2017

I didn’t say anything about an outside force guiding us. I am saying that if the structure of reality has characteristics in which certain moral values produce evolutionary successful outcomes, it follows that these moral values correspond to an objective evolutionary reality

I agree that this is a perfectly fine way to think of things. We may not disagree on any factual questions.

Here’s a factual question some people get tripped up by: would any sufficiently intelligent being rediscover and be motivated by a morality that looks a lot like human morality? Like, suppose there was a race of aliens that evolved intelligence without knowing their kin—would we expect them to be motivated by filial love, once we explained it to them and gave them technology to track down their relatives? Would a superinntelligent AI never destroy humans, because superintelligence implies an understanding of the value of life?

Would it be OK to enslave half of humanity...

No. Why? Because I would prefer not. Isn’t that all that can be sufficient to motivate my decision? A little glib, I know, but I really don’t see this as a hard question.

When people say “what is right?, I always think of this as being like “by what standard would we act, if we could choose standards for ourselves?” rather than like “what does the external rightness-object say?”

We can think as if we’re consulting the rightness-object when working cooperatively with other humans—it will make no difference. But when people disagree, the approximation breaks down, and it becomes counter-productive to think you have access to The Truth. When people disagree about the morality of abortion, it’s not that (at least) one of them is factually mistaken about the rightness-object, they are disagreeing about which standard to use for acting.

Manfred Jun 29, 2017, 3:27 PM
0 points
in reply to: Erfeyah’s comment on: Open thread, June 26 - July 2, 2017
Human morals, human preferences, and human ability to work to satisfy those morals and preferences on large scales, are all quite successful from an evolutionary perspective, and make use of elements seen other places in the animal kingdom. There’s no necessity for any sort of outside force guiding human evolution, or any pre-existing thing it’s trying to mimic, therefore we shouldn’t presume one.

Let me give an analogy for why I think this doesn’t remove meaning from things (it will also be helpful if you’ve read the article Fake Reductionism from the archives). We like to drink water, and think it’s wet. Then we learn that water is made of molecules, which are made of atoms, etc, and in fact this idea of “water” is not fundamental within the laws of physics. Does this remove meaning from wetness, and from thirst?

Manfred Jun 29, 2017, 12:38 AM
0 points
in reply to: Erfeyah’s comment on: Open thread, June 26 - July 2, 2017
I think this is totally consistent with relativism (Where I mean relativism in the sense of moral and aesthetic judgments being based on subjective personal taste and contingent learned behavior. Moral and aesthetic judgments still exist and have meaning.).

The fact that people make the same moral judgments most of the time is (I claim) because humans are in general really, really similar to each other. 200 years ago this would be mysterious and might be taken as evidence of moral truths external to any human mind, but now we can explain this similarity in terms of the origin of human values by evolution.

Manfred Jun 26, 2017, 8:57 PM
0 points
in reply to: Thomas’s comment on: Open thread, June 26 - July 2, 2017
To detect tides with a symmetrical wheel, start it spinning. It will experience a gyroscopic torque due to tidal forces. This is actually the same mechanism that causes the earth’s axis to precess.

Manfred Jun 22, 2017, 11:53 PM
1 point
in reply to: cousin_it’s comment on: Open thread, June. 19 - June. 25, 2017
Do you intend “‘Surprise implies Mon’ is false” to mean “Not (Surprise and Mon)”? I’m just a little confused because I think in classical logic, if Surprise is false then all the implication statements are true. Therefore if at least one of Mon-Fri is true, Surprise cannot be false. Maybe I should have read the paper you reference.

Personally, I think the original presentation is already sufficient. If the judge really refuses to hang the prisoner if they’re not surprised, and the judge is in some sense transparent to the prisoner, so that the prisoner knows what logic they will follow, then the judge will not hang the prisoner.

The paradox is in the punchline that if the judge then hangs the prisoner on wednesday, the prisoner is surprised. But this is simply because it’s a counterfactual that won’t happen, given the toy model of how the judge works. If the prisoner merely has a bad model of the judge and therefore makes wrong predictions, this isn’t much of a paradox, though still a decent joke.

I think this problem is clearer if we imagine a game where the judge outputs a probability distribution over which day the prisoner will be executed, and the prisoner is postulated to be clever enough to figure out what probability distribution the judge will output.

In this scenario, it is very easy for the judge to surprise the prisoner most of the time—a uniform distribution will suffice. If the judge’s goal is to minimize the expected information the prisoner has about whether they’ll die, on the day of the execution, then maybe you get some slightly more complicated probability distribution. But within this game, it is impossible for the judge to output a probability distribution such that the prisoner is always surprised, never unsurprised. Every probability distribution supported on Mon-Fri has a final day, on which the prisoner is not surprised.

So if the judge says they’re going to output a probability distribution such that the prisoner will definitely be surprised when they die, they are telling a lie.

Manfred Jun 8, 2017, 6:58 PM
2 points
on: Destroying the Utility Monster—An Alternative Formation of Utility
Interpersonal utility comparisons are not a natural part of utility theory the same way individual utility functions are. I think of them as being important for two reasons:

1: Our own personal moral reasoning does something like interpersonal utility comparisons, and we want to try to formalize that.

2: we want to cooperate with other people to achieve some goal that will benefit all, but in order to define our collective goals we need some form of interpersonal utility comparison.

2.5: We’re about to build an AI and want to program in by hand how it should weigh human values (warning: don’t do 2.5).