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(As suggested on the comment, I slightly changed my previous question.)

I have $N$ coins and I am testing them one by one if it is fair or not. I know that, if it is unfair, the probability of head will be 2/3.

Given that $n=10$, for instance, how can I express the Bayesian updating process of my continuous prior $p_{0}$? Would the posterior converge to the true distribution even if $n$ is left to be a small number?

Thank you very much.

김찬우
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  • "I believe that $p_0$ proportion of coins are unfair". Taken literally, this statement means you believe *a priori* that exactly $p_0 N$ out of the $N$ coins are unfair, and that all other numbers are impossible. Bayesian inference can't update this kind of 'certain' belief; it will remain fixed despite all observed evidence. But perhaps you meant something else (please edit the question if so). – user20160 Mar 04 '21 at 13:11
  • A couple possibilities: 1) Specify a non-degenerate prior over the proportion of unfair coins (which must be a multiple of $\frac{1}{N}$), or the number of them (which might be a little more convenient). Note that this is a probability distribution over all possible values, not just a single number. 2) Declare that each coin has an independent prior probability $p_0$ of being unfair. This is like imagining that the $N$ coins are randomly drawn from a giant bag of coins, where $p_0$ is the fraction of unfair coins in the bag. – user20160 Mar 04 '21 at 13:11
  • Here I think the first interpretation is more what I meant to say. Thank you very much! – 김찬우 Mar 04 '21 at 13:45
  • So, if I understand well the $p_{0}$ can be interpreted as the probability of a coin being unfair? – Fiodor1234 Mar 04 '21 at 13:59
  • no indeed I wanted to say that $p_{0}$ is the prior of the number of unfair coins among N coins. – 김찬우 Mar 04 '21 at 14:03
  • @김찬우 You said in the comments that $p_0$ is a prior on the proportion of unfair coins, but the question still defines $p_0$ as a single number. If it's a prior, you must define a probability distribution over possible proportions – user20160 Mar 04 '21 at 14:12
  • Ah now I get your point. So essentially in this problem, the prior is continuous and the prior cannot be degenerate prior as a point mass at a certain value. Is that so? – 김찬우 Mar 04 '21 at 14:20
  • @김찬우 Well, the prior should actually be discrete, because the only valid proportions are integer multiples of $\frac{1}{N}$. But, yes, if you want to learn anything from the data, it cannot be a point mass. – user20160 Mar 04 '21 at 14:29
  • The second option I mentioned above might be a bit easier to work with – user20160 Mar 04 '21 at 14:35
  • Thank you very much! I am not exactly working with data. Anyway, When $N \rightarrow \infty $, do the first interpretation and the second go to the same limit distribution? If so, could you right an answer based on the second interpretation? – 김찬우 Mar 04 '21 at 14:42
  • Not sure offhand. I can try to take a crack at it later. Just one last note: your final edit to the question doesn't specify what quantity you want to compute a posterior over--it's the number or proportion of unfair coins, right? – user20160 Mar 04 '21 at 14:50
  • Yest it is. Just one more quick question. If I add a new constraint that if I stop flipping a coin that turns tail in a row, would that hurt the conclusion that the posterior converges to "true" distribution? Thank you very much. I learnt a lot through your comments ! – 김찬우 Mar 04 '21 at 15:38

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