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The conjugate prior of the multinomial distribution is Dirichlet -- it is a distribution over the parameters (the probabilities of outcomes) of the multinomial.

Define

D = Dirichlet(X + 1)


(the 1 represents the non-informative prior belief that all outcomes probability vectors are equiprobable.)

And then integrate over the region of the pdf that you're interested in.

I can expand on this answer if you like.

1

The conjugate prior of the multinomial distribution is Dirichlet -- it is a distribution over the parameters (the probabilities of outcomes) of the multinomial.

Define

D = Dirichlet(X + 1)


(the 1 represents the non-informative prior belief that all outcomes are equiprobable.)

And then integrate over the region of the pdf that you're interested in.

I can expand on this answer if you like.