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Given two coins with respective biases $\mu_a$ and $\mu_b$, suppose that we have no information on their biases, but we believe that the two biases are identical or similar. Is there a standard/natural Bayesian way to encode this belief?

One possible approach I can think of is to do this hierarchically, something like:

$\mu_z\sim Uniform(0,1)$

$\sigma\sim \ldots$

$\mu_a \sim TruncatedNormal(mu=\mu_z, \sigma, range=[0,1])$ $\mu_b \sim TruncatedNormal(mu=\mu_z, \sigma, range=[0,1])$

Ideally, it would be great to have something that yields an estimator for e.g. $\mu_a$ that is a weighted combination of the MLE for $\mu_a$ and the MLE for the overall $\frac{\#heads}{\#flips}$ of the two coins together, something like:

$\hat{\mu}_{a} = \frac{\#heads_a + \sigma \cdot \frac{\#heads_a + \#heads_b}{\#flips_a + \#flips_b}}{\#flips_a + \pi}$ where $\sigma$ is a prior parameter that captures the strength of our belief that the two coins' biases are similar.

But is there a standard Bayesian approach here, or some distributions that are typically used?

cataclysmic
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    "identical or similar" is confusing. If the two parameters are _identical_ they are one parameter. If they are _similar_, some information must be provided on the meaning of "similar" to chose a prior on their difference (or their similarity). – Xi'an May 05 '20 at 08:43
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    In https://stats.stackexchange.com/questions/87358/multivariate-beta-distribution-no-dirichlet they are referring to a paper dealing with a 2d beta distribution. I think this is something that you should look at: There should be a parameter telling the 2d-beta-distribution how 'narrow' it is distributed along the diagonal. If the belief is 0 then it should collapse to unif([0,1]x[0,1]) and if the belief is 1 then it should collapse to a 1d beta distribution going perfectly along the diagonal... – Fabian Werner May 05 '20 at 08:48
  • @Xi'an, I would be interested in 'standard' approaches that capture either the idea that $\mu_a$ and $\mu_b$ are 'identical with some probability', or alternatively, that they are 'similar' i.e. Pr[$\mu_a - \mu_b = x$] is shrinking in $|x|$. Basically, I have some freedom to define the model here, and I'm trying to find out what standard approaches for similar problems are. – cataclysmic May 05 '20 at 22:09
  • @Fabian Thank you for this link---this is very apt and shows an approach I had not considered. Obviously, my 'suggested estimator' in the question is just derived from a reparametrized Beta conjugate prior, but the 2D beta with a 'similarity parameter' is a very interesting idea. – cataclysmic May 05 '20 at 22:10

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