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In their paper published here, Zoubin and Griffith present an introduction to Indian Buffet Process. (If you are interested in the topic, is a must read) My question is quite simple. In page 1186 in equation 2. They define the Dirichlet density to be equal to the marginal of the multinomial over the simplex. How could they define this?, since the dirichlet is a multinomial times the inverse of the beta function. Is this because they are marginalizing the Dirichlet over the theta parameters? I'm a bit confused by this. Thanks |
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I'm not sure what you're question is, so I'll answer three possibilities. I'm surprised they used the letter D and called the Dirichlet normalizing constant instead of its more common name: the multinomial beta function. Is your question that they chose to call it D instead of B? It's the same thing. Is your question about the provenance of the Dirichlet density? You can derive the Dirichlet density without the normalization constant by a straightforward manipulation of the multinomial density, e.g., see this link. If you are asking why does the normalization constant work out to be the multinomial Beta function, there's a very pretty proof for the Beta distribution here and for the Dirichlet distribution here. (Note that the parameters in both solutions are offset by one from the usual Beta/Dirichlet distribution parameters.) |
(hand-waving) When you integrate out the multinomials over the simplex you get the inverse beta function, which is the normalizing constant of the dirichlet, as it does not depend on the probabilities, just on the alphas.
Equation 2 is simply the normalizing constant for a Dirichlet distribution with parameters alpha_1,ldots,alpha_K. It is not a distribution; it is the normalizing constant for the appropriate distribution over thetas.
I'm sure there's some fancy math way to show that the integral is equal to the ratio of the product of gamma functions over the gamma function of the sum, but it's not really necessary to understand how to use the IBP.
That's the fancy math that is pretty nice to know so we can do extensions :)
You're probably best off looking into real analysis then. I've seen books on integrals involving the Gamma function. Almost all the stat treatments I've seen just show that it is the solution to the integral. Good luck!
@Leon: Can you clarify your question or mark my answer as correct?