## How to update beliefs about correlated categorial distributions

I will be making one observation at a time from one of many correlated distributions and each observation will fall into one of four categories.

If $p_i$$_j$ is the prior belief about the probability that i is observed from distribution j, what are posterior probabilities if i* is observed from distribution j*? Also, I am unsure what my covariance matrix would look like or how I would update it.

The only paper I have found that seems to be related is http://sankhya.isical.ac.in/search/63b3/8436fnl.pdf, but I had trouble following the notation.

Motivation:

Three prices are chosen: price of good 1, price of good 2, price of buying both. Each customer makes a decision (1, 2, both, or neither) and then a manager needs to update beliefs about the probability of each choice. I'll make the price choices discrete and make the prior correlation dependent on the distance between price vectors.

I think I already have a good idea of how I would like to simulate customer arrivals and the decision process.

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 "Make the prior correlation dependent on the distance between price vectors" is vague. A convenient prior for categorical distributions is the conjugate prior known as the dirichlet prior. You can model your distribution parameters as i.i.d samples from a dirichlet distribution. You now need a prior over your dirichlet parameter. There is a conjugate prior for the dirichlet distribution, but it seems obscure (see discussion here: andrewgelman.com/2009/04/conjugate_prior). You can simply use the conjugate prior of the gamma distribution for each dirichlet parameter as Gelman suggests. – Arthur B Apr 9 2012 at 13:35