Integral of a Negative Multinomial in a subset space

by alberto   Last Updated March 07, 2017 15:19 PM

I have a Negative Multinomial distribution, defined as:

\begin{align} P(k_1,...k_D | k_0, \mathbf{p}) = \frac{\Gamma(k_0 + \sum_i k_i) }{\Gamma(\alpha)\prod_{i} k_{i}!} p_0^{k_0} \prod_{i=1}^D p_i^{k_{i}} \end{align}

The integral over all possible count vectors $k_1,...k_D$ is 1 by definition, since this is a valid probability distribution.

I want to compute the sum over some pre-computed subset of count vectors $\mathcal{K}$:

\begin{align} \sum_{\mathbf{k} \in \mathcal{K}} P(k_1,...k_D | k_0, \mathbf{p}) = \sum_{\mathbf{k} \in \mathcal{K}} \frac{\Gamma(k_0 + \sum_i k_i) }{\Gamma(\alpha)\prod_{i} k_{i}!} p_0^{k_0} \prod_{i=1}^D p_i^{k_{i}} \end{align}

How can I do it? I guess, for instance, I could get some insight from seeing the demonstration of the integration over all possible $\mathbf{k}$ (if such demonstration exists) and then tweaking it to sum through my subspace?

Or maybe it can't be done analytically?

My final goal is to find the MLE estimate of $\mathbf{p}$ for that subset $\mathcal{K}$.

Any answer or reference is very welcome.

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