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I have a N*M matrix with N customers and M products. Each row of this matrix is a M dimensional vector like [1 3 4 1 5 ....] where each value represents how many times that customer has chosen this product in the past. Given this data, I want to predict this M dimensional vector for each customer in the future.

This prediction should be such that, for each customer, the M length vector should sum to 1(the vector should hold normalized probabilities for each product being clicked. For ex- [0.99 0 0 .. 0 0.01] for customer i, says that the product 1 has the highest probability of being chosen and product M has 0.01 probability of being chosen and others have 0 probability).

How do I apply bayesian inference to this problem? I can do MLE by dividing the counts in each vector by its total(for ex- for customer i, his product vector can be ([1 3 4 1 5 ....] / sum([1 3 4 1 5 ....]) and this will be normalized). I want to know how to apply bayesian inference to solve this problem?

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I think your problem of 'predicting' the matrix in the future can be rephrased as a recommendation problem: you are trying to recommend (predict) products for users, i.e. you are trying to tell what will they click on.

People usually don't use Bayesian methods in recommendation (because other methods are better suited for this kind of problem) but maybe you can check out this article:

http://www.ics.uci.edu/~pazzani/Publications/IPSJ.pdf

The authors here use a Bayes classifier to do collaborative filtering (which is a form of recommendation).

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