A distribution similar to a Negative Binomial of order k

Question

Given an integer observation $v$, I consider it comes from a Poisson distribution centered at $\boldsymbol{\theta}^T \boldsymbol{\lambda}$ where $\boldsymbol{\theta, \lambda}$ are K-dimensional integer vectors.

\begin{align} v \sim Poisson(\sum_k\theta_k\lambda_k) & \quad\text{ or }\quad v = \sum_kPoisson(\theta_k\lambda_k)\\ \lambda &\sim Gamma (\alpha, \beta) \end{align}

I we marginalize over $\lambda$, we get the following generative model:

\begin{align} v &= \sum_k NegativeBinomial(\alpha, p_k = \frac{\theta_k}{\beta + \theta_{k}}) \end{align}

and the probability distribution gives: \begin{align} p(v | \boldsymbol{P})=& \sum_{c_1 + ... c_K = v} \prod_k \left[ \frac{\Gamma(\alpha_k + c_{k})}{\Gamma(\alpha)c_{k}!} (1 - p_{k})^{\alpha} p_{k}^{c_{k}} \right] \end{align}

where the sum is over all integer partitions (or compositions, since $c_i = 0$ is allowed) of $v$: $c_1+...+c_k = v$

Looking for some similar distribution, I found the "Negative Binomial of order $k$". However, the partitions of the Negative Binomial of order $k$ are such that \begin{align} \sum_k kc_k = v \end{align}

Does my probability distribution correspond (or is similar) to some known distribution?

Answer 1

The distribution of $\nu$ can be derived in a slightly different way. We can write $\nu = \sum_k \nu_k$, $\nu_k \sim \text{Poisson}(\theta_k \lambda_k)$. Each product $\theta_k \lambda_k$ has (let us say; it depends on the parameterization of the Gamma) distribution $\text{Gamma}(\alpha, \beta \theta_k)$. Integrating out $\lambda_k$ gives us a Negative Binomial distribution for each of the $\nu_k$ with parameters $\alpha, \frac{\beta\theta_k}{1 + \beta\theta_k}$. Thus, the distribution of $\nu$ is that of the sum of $K$ negative binomial variates with shape parameter $\alpha$ and probability parameters $\frac{\beta\theta_k}{1 + \beta\theta_k}$.

Unfortunately, this distribution does not have a tractable form (unless all the $\theta_k$ are equal,) nor does it have a name, as far as I know.