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On the evidence of a single coin toss

blog.moertel.com

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Re: On the evidence of a single coin toss

#11

Post by hammock »

blahedo wrote: post #10

There are at least three different lines of inquiry here: - Hypothesis testing. If the [null] hypothesis is that p(heads) is 1, you can't prove this, only disprove it. So: "doesn't sway". Not very interesting, but there it is. - Simple Bayesian. The probability of his claim given that it comes up heads, p(C|H), is the prior of his claim, p(C), times p(H|C), divided by p(H). Well, p(H|C) is 1 (that is the claim), and…

This answer makes the most sense to me, though I dont understand it.

Some quantity x which is your trust in Tom, plus some quantity y which is the probability of the coin multiplied by some factor of increasing confidence

Re: On the evidence of a single coin toss

#12

Post by hackerblues »

blahedo wrote: post #10

There are at least three different lines of inquiry here: - Hypothesis testing. If the [null] hypothesis is that p(heads) is 1, you can't prove this, only disprove it. So: "doesn't sway". Not very interesting, but there it is. - Simple Bayesian. The probability of his claim given that it comes up heads, p(C|H), is the prior of his claim, p(C), times p(H|C), divided by p(H). Well, p(H|C) is 1 (that is the claim), and…

For a different approach: from a scientific/logical perspective the result has no bearing on the truth of his claim. The only possibility which we can remove from consideration is the claim that "The coin will always land tails."

Re: On the evidence of a single coin toss

#13

Post by jules »

Jabbles wrote: post #8
jules wrote: post #5

Say you have a prior probability distribution P(p) for the probability you think the coin is a coin that comes up heads with probability p. Your probability distribution P(p) will probably have a huge peak around p=0.5, but you can choose any prior belief. So P(p) is your opinion about the coin prior to seeing the experiment. Now we can apply Bayes' theorem to compute your opinion P'(p) about the coin after seeing th…

Your prior belief that the coin is "special" should be extremely small, since you've examined it and can see no reason for it to come up heads all the time. After flipping, your belief will be larger, but still small.

Any prior belief is valid. That's the point of this: you separate the mathematical reasoning from the subjective assumptions (beliefs). For example it might be the case that Tom has demonstrated several of those special coins before, and in that case your opinion would probably be that there is a good chance that this one is special too. The nice thing about the math is that we can encapsulate these assumptions in the prior probability distribution P(p).

BTW I used that distribution in the plots because it was easiest to come up with, and somewhat realistic, and it shows the skewing well. Feel free to plug in your own beliefs.

Re: On the evidence of a single coin toss

#14

Post by khafra »

blahedo wrote: post #10

There are at least three different lines of inquiry here: - Hypothesis testing. If the [null] hypothesis is that p(heads) is 1, you can't prove this, only disprove it. So: "doesn't sway". Not very interesting, but there it is. - Simple Bayesian. The probability of his claim given that it comes up heads, p(C|H), is the prior of his claim, p(C), times p(H|C), divided by p(H). Well, p(H|C) is 1 (that is the claim), and…

Nice walkthrough of bayesian probability, then bayesian epistemology. All that's missing is a link to http://yudkowsky.net/rational/technical for those who'd like furhter readings on the subject.

Re: On the evidence of a single coin toss

#15

Post by blahedo »

blahedo wrote: post #10

There are at least three different lines of inquiry here: - Hypothesis testing. If the [null] hypothesis is that p(heads) is 1, you can't prove this, only disprove it. So: "doesn't sway". Not very interesting, but there it is. - Simple Bayesian. The probability of his claim given that it comes up heads, p(C|H), is the prior of his claim, p(C), times p(H|C), divided by p(H). Well, p(H|C) is 1 (that is the claim), and…

A little dense, sorry; I reposted on my blog with slightly better formatting (and at least breaking out some of the math onto separate lines):

http://www.blahedo.org/blog/archives/001081.html

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