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Beautiful Probability

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Re: Beautiful Probability

#21

Post by lalaithion »

AbrahamParangi wrote: post #18

I'm confused in that I don't see how this is troubling. Yes, the two experimenters rolled dice and got the same result, but it's as if one of them was rolling a 6 sided die and the other a 20 sided one. Each experiment is not a result per se but a sample from a distribution. How you infer the shape of that distribution based on the experiment is a function of the distribution of all courses your experiment could have…

If you see two people roll a d20 and get a 20, you get to say "wow, that was unlikely" to both of them, even if one of them privately admits they were going to quickly re-roll their die if they got below a 10. What matters is their actual behavior (identical in the example) not their intentions. The d6 vs d20 version is different because their behavior is different.

Re: Beautiful Probability

#22

Post by birdofhermes »

As other commenters have pointed out any given introductory chapter in a book on Bayesian statistics, including Jaynes’, is better exposition than this. I found _Probability Theory: The Logic of Science_ very easy to follow and very well-written.

I had a similar experience when I finally found a copy of Barbour’s _The End of Time_ and discovered, much to my chagrin, that it wasn’t nearly as mystical or complicated as EY makes it seem in the Timeless Physics “sequence”. Barbour’s account was much more readable and much easier to understand.

Yudkowsky just isn’t that great of a popular science writer. It’s not his specialty, so this shouldn’t be surprising.

Re: Beautiful Probability

#23

Post by ninthcat »

lalaithion wrote: post #21
AbrahamParangi wrote: post #18

I'm confused in that I don't see how this is troubling. Yes, the two experimenters rolled dice and got the same result, but it's as if one of them was rolling a 6 sided die and the other a 20 sided one. Each experiment is not a result per se but a sample from a distribution. How you infer the shape of that distribution based on the experiment is a function of the distribution of all courses your experiment could have…

If you see two people roll a d20 and get a 20, you get to say "wow, that was unlikely" to both of them, even if one of them privately admits they were going to quickly re-roll their die if they got below a 10. What matters is their actual behavior (identical in the example) not their intentions. The d6 vs d20 version is different because their behavior is different.

Unlikely in what probability space? We only see one version of reality so the probabilities that we assign to any outcome are based on a prior choice of probability space. That is why the researchers' intent matters.

Re: Beautiful Probability

#24

Post by AbrahamParangi »

ninthcat wrote: post #23
lalaithion wrote: post #21

Earlier quoted context omitted.

If you see two people roll a d20 and get a 20, you get to say "wow, that was unlikely" to both of them, even if one of them privately admits they were going to quickly re-roll their die if they got below a 10. What matters is their actual behavior (identical in the example) not their intentions. The d6 vs d20 version is different because their behavior is different.

Unlikely in what probability space? We only see one version of reality so the probabilities that we assign to any outcome are based on a prior choice of probability space. That is why the researchers' intent matters.

Yes, indeed.

Re: Beautiful Probability

#25

Post by AbrahamParangi »

lalaithion wrote: post #21
AbrahamParangi wrote: post #18

I'm confused in that I don't see how this is troubling. Yes, the two experimenters rolled dice and got the same result, but it's as if one of them was rolling a 6 sided die and the other a 20 sided one. Each experiment is not a result per se but a sample from a distribution. How you infer the shape of that distribution based on the experiment is a function of the distribution of all courses your experiment could have…

If you see two people roll a d20 and get a 20, you get to say "wow, that was unlikely" to both of them, even if one of them privately admits they were going to quickly re-roll their die if they got below a 10. What matters is their actual behavior (identical in the example) not their intentions. The d6 vs d20 version is different because their behavior is different.

Let's imagine that we ran it as a simulation and we ran it a million times. The two people would have a different distribution of results. If you ignore the intention, you ignore reality as if that intention were not a part of it.

Do you not notice that your inference is less accurate using this line of reasoning? Does that not suggest that it's simply wrong?

Re: Beautiful Probability

#26

Post by usgroup »

lalaithion wrote: post #16
usgroup wrote: post #5

So you know when you believe something and then you update your belief because you get some evidence? Yeah, and then you stack some beliefs on top of that. And then you discover the evidence wasn’t actually true. Remind me again what the normative Bayesian update looks like in that instance. Unfortunately it’s turtles all the way down.

P(B|I saw E, P) = P(I saw E|B,P) * P(B|P) / P(I saw E|P) P(B|E was false, I saw E, P) = P(E was false|B,I saw E,P) * P(B|P,I saw E) / P(E was false|P, I saw E) This is a pretty basic application of Bayes' theorem.

Love it: p(I saw E) and p(I didn’t really see E).

Just move the argument one level down: “I saw E is false” and it turns out so is “E is false” . So then? Add “E was false was false”?

Turtles all the way down.

At some point something has to be “true” in order to conditionalise on it.

Re: Beautiful Probability

#27

Post by usgroup »

AbrahamParangi wrote: post #25
lalaithion wrote: post #21

Earlier quoted context omitted.

If you see two people roll a d20 and get a 20, you get to say "wow, that was unlikely" to both of them, even if one of them privately admits they were going to quickly re-roll their die if they got below a 10. What matters is their actual behavior (identical in the example) not their intentions. The d6 vs d20 version is different because their behavior is different.

Let's imagine that we ran it as a simulation and we ran it a million times. The two people would have a different distribution of results. If you ignore the intention, you ignore reality as if that intention were not a part of it. Do you not notice that your inference is less accurate using this line of reasoning? Does that not suggest that it's simply wrong?

This is well put. Coincidentally in the example the results are the same , but they need not be. given repeated experiments with the same intentions one may expect different distributions.

However, one could just move the argument up a level and manufacture a case of different intentions leading to the same distributions and then ask the same question.

Re: Beautiful Probability

#28

Post by kgwgk »

AbrahamParangi wrote: post #18

I'm confused in that I don't see how this is troubling. Yes, the two experimenters rolled dice and got the same result, but it's as if one of them was rolling a 6 sided die and the other a 20 sided one. Each experiment is not a result per se but a sample from a distribution. How you infer the shape of that distribution based on the experiment is a function of the distribution of all courses your experiment could have…

The question is whether we should draw different conclusions from one set of observations depending not just on what we are observing but also on different ways to define "what are all the worlds that I might be in that are consistent with what I'm presently observing".

Re: Beautiful Probability

#29

Post by kevindamm »

lalaithion wrote: post #17
d0mine wrote: post #4

Bayesian approach sounds like a religion (one true way). There is nothing unusual about different mathematical methods/models producing different results e.g., the number of roots even for the same quadratic equation may depend on "private" thoughts such as whether complex roots are of interest (sometimes they do/sometimes they don't). All models are wrong some are useful.

> the number of roots even for the same quadratic equation may depend on "private" thoughts such as whether complex roots are of interest You are confusing ambiguity in a problem statement due to human language being imprecise with two well-specified identical experimental results having different results due to the intentions of the human carrying them out. Is arithmetic a religion because there's "one true way" of…

I can think of at least two ways to add integers.. the categorical way that applies a mapping from the set into itself, and the set-theoretic way that deals with unwrapping and rewrapping successor relations. The latter is sometimes resorted to in heavily-relational contexts like Datalog.

Re: Beautiful Probability

#30

Post by lalaithion »

ninthcat wrote: post #23
lalaithion wrote: post #21

Earlier quoted context omitted.

If you see two people roll a d20 and get a 20, you get to say "wow, that was unlikely" to both of them, even if one of them privately admits they were going to quickly re-roll their die if they got below a 10. What matters is their actual behavior (identical in the example) not their intentions. The d6 vs d20 version is different because their behavior is different.

Unlikely in what probability space? We only see one version of reality so the probabilities that we assign to any outcome are based on a prior choice of probability space. That is why the researchers' intent matters.

Both events have the same probability of happening; 1/20. The fact that the researcher intended to do something in a reality that didn't happen isn't relevabnt.
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