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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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