# How to calculate a Bayesian Inference over a Poisson Binomial Distribution

In relation to this question, how do I use Bayesian inference over a Poisson Binomial Distribution? If possible, what is the Conjugate Prior?

Thanks to @Stijn, here is an elaboration of the problem:

Consider a group of 5 people each with a Red, Blue, Yellow or Green ball in their pocket. Let $R_p, B_p, Y_p \text{ and } G_p$ be the probability that person $p$ has that colour ball in their pocket. Obviously $R_p + B_p + Y_p + G_p=1$.

You are allowed to guess what each person has in their pocket. Assume that for each person you guess $\max(R_p, B_p, Y_p, G_p)$. You are then told how many guesses you got correct $k=\{0,1,2,3,4,5\}$ but not which guesses were correct.

Given the new piece of information $k$, how do you revise the estimates of $R_p, B_p, Y_p \text{ and } G_p$.

Obviously, if $k=5$ then $\max(R_p, B_p, Y_p, G_p)=1$ and $not \max(R_p, B_p, Y_p, G_p)=0$ and if $k=0$ then $\max(R_p, B_p, Y_p, G_p)=1$ and $not \max(R_p, B_p, Y_p, G_p)=\frac{not \max(R_p, B_p, Y_p, G_p)}{1-\max(R_p, B_p, Y_p, G_p)}$

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You don't necessarily need a conjugate prior to do Bayesian inference. What is it exactly that you attempt to model? –  Stijn Jun 26 '13 at 8:55