# Sum of i.i.d Negative Multinomials

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}

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?

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.

I think using MLE here may give you misleading conclusions. You know that some outcome in $\mathcal{K}$ has occured and you want to compute and maximise the likelihood $L(\mathbf{p})=P(k\in\mathcal{K})$. For example, suppose the experiment is stopped when $k_0=1$ events in the category 0 has occured and the number of events in the other categories were any $$k\in \mathcal{K}=\{(10,0,0,0),(0,0,0,10)\}$$ This tells you that $\mathbf{p}$ must be either quite close to $\mathbf{p}=(1/11,10/11,0,0,0\}$ or $\mathbf{p}=(1/11,0,0,0,10/11\}$ and this will appear as two optima in $L(\mathbf p)$ close to these values.
You may be better off doing Bayesian inference, perhaps with a Dirichlet prior on $\mathbf{p}$. The resulting posterior would then be a Dirichlet mixture with components associated with each element in $\mathcal{K}$.
• Ok, I guess the likelihood may be unimodal if all elements of $\mathcal{K}$ are clustered together, in which case MLE may give reasonable results. If so, I don't really see where the difficulty with computing the MLEs of $\mathbf{p}$ (numerically by brute force), but perhaps an analytic expression is available if $\mathcal{K}$ has a specific form? Mar 13, 2017 at 13:00