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I'm trying to compute the individual posterior probabilities from my estimated multinomial probit model. I have obtained the latent utilities as well as the posterior parameters for beta and sigma for each individual, but now I am confused how to obtain the posterior probabilities.

For the binomial probit model it seems straightforward to obtain the posterior probabilities by calculating:

E[Φ(x'beta]

for each individual, where Φ represents the cumulative normal distribution . But I don't understand how to translate this to the multinomial scenario.

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The Multinomial's equivalent of the prior Beta distribution is the prior Dirichlet distribution.

You can think of the concentration parameters of the posterior Dirichlet as essentially a vector of posterior 'Beta-ish' probabilities of each corresponding class, so you can apply the same procedure you're already familiar with.

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