Dirichlet negative multinomial distribution

The Dirichlet distribution is a conjugate distribution to the negative multinomial distribution. This fact leads to an analytically tractable compound distribution.

For a random vector of category counts $\backslash mathbf\{x\}=(x\_1,\backslash dots,x\_m)$, distributed according to a negative multinomial distribution, the compound distribution is obtained by integrating on the distribution for p which can be thought of as a random vector following a Dirichlet distribution:

:$\backslash Pr(\backslash mathbf\{x\}\backslash mid\; x\_0,\; \backslash alpha\_0,\; \backslash boldsymbol\{\backslash alpha\})=\backslash int\_\{\backslash mathbf\{p\}\}\backslash mathrm\{NegMult\}(\backslash mathbf\{x\}\backslash mid\; x\_0,\; \backslash mathbf\{p\})\; \backslash mathrm\{Dir\}(\backslash mathbf\{p\}\backslash mid\backslash alpha\_0,\backslash boldsymbol\{\backslash alpha\})\backslash textrm\{d\}\backslash mathbf\{p\}$

:$\backslash Pr(\backslash mathbf\{x\}\backslash mid\; x\_0,\; \backslash alpha\_0,\; \backslash boldsymbol\{\backslash alpha\})=\{\backslash frac\{\backslash Gamma\backslash left(\backslash sum\_\{i=0\}^m\{x\_i\}\backslash right)\}\{\backslash Gamma(x\_0)\backslash prod\_\{i=1\}^m\; x\_i!\}\}\; \backslash frac\{1\}\{\backslash mathrm\{B\}(\backslash boldsymbol\backslash alpha\_+)\}\backslash int\_\{\backslash mathbf\{p\}\}\; \backslash prod\_\{i=0\}^m\; p\_i^\{x\_i+\backslash alpha\_i\; -\; 1\}\backslash textrm\{d\}\backslash mathbf\{p\}$

which results in the following formula:

:$\backslash Pr(\backslash mathbf\{x\}\backslash mid\; x\_0,\; \backslash alpha\_0,\; \backslash boldsymbol\{\backslash alpha\})=\{\backslash frac\{\backslash Gamma\backslash left(\backslash sum\_\{i=0\}^m\{x\_i\}\backslash right)\}\{\backslash Gamma(x\_0)\backslash prod\_\{i=1\}^m\; x\_i!\}\}\; \backslash frac\{\{\backslash mathrm\{B\}\}(\backslash mathbf\{x\_+\}+\backslash boldsymbol\backslash alpha\_+)\}\{\backslash mathrm\{B\}(\backslash boldsymbol\backslash alpha\_+)\}$

where $\backslash mathbf\{x\_+\}$ and $\backslash boldsymbol\backslash alpha\_+$ are the $m+1$ dimensional vectors created by appending the scalars $x\_0$ and $\backslash alpha\_0$ to the $m$ dimensional vectors $\backslash mathbf\{x\}$ and $\backslash boldsymbol\backslash alpha$ respectively and $\backslash mathrm\{B\}$ is the multivariate version of the beta function. We can write this equation explicitly as

:$\backslash Pr(\backslash mathbf\{x\}\backslash mid\; x\_0,\; \backslash alpha\_0,\; \backslash boldsymbol\{\backslash alpha\})=x\_0\backslash frac\{\backslash Gamma(\backslash sum\_\{i=0\}^m\; x\_i)\backslash Gamma(\backslash sum\_\{i=0\}^m\; \backslash alpha\_i)\}\{\backslash Gamma(\backslash sum\_\{i=0\}^m\; (x\_i+\backslash alpha\_i))\}\; \backslash prod\_\{i=0\}^m\; \backslash frac\{\backslash Gamma(x\_i+\backslash alpha\_i)\}\{\backslash Gamma(x\_i+1)\backslash Gamma(\backslash alpha\_i)\}.$

Alternative formulations exist. One convenient representationFarewell, Daniel & Farewell, Vernon. (2012). Dirichlet negative multinomial regression for overdispersed correlated count data. Biostatistics (Oxford, England). 14. 10.1093/biostatistics/kxs050. is

:$\backslash Pr(\backslash mathbf\{x\}\backslash mid\; x\_0,\; \backslash alpha\_0,\; \backslash boldsymbol\{\backslash alpha\})=\; \backslash frac\{\backslash Gamma(x\_\backslash bullet)\}\{\backslash Gamma(x\_0)\backslash prod\_\{i=1\}^m\; \backslash Gamma(x\_i+1)\}\; \backslash times\; \backslash frac\{\backslash Gamma(\backslash alpha\_\backslash bullet)\}\{\backslash prod\_\{i=0\}^m\; \backslash Gamma(\backslash alpha\_i)\}\; \backslash times\; \backslash frac\{\backslash prod\_\{i=0\}^m\; \backslash Gamma(x\_i+\backslash alpha\_i)\}\{\backslash Gamma(x\_\backslash bullet+\backslash alpha\_\backslash bullet)\}$

where $x\_\backslash bullet=\; x\_0+x\_1+\; \backslash cdots\; +\; x\_m$ and $\backslash alpha\_\{\backslash bullet\}=\; \backslash alpha\_0+\backslash alpha\_1+\; \backslash cdots\; +\; \backslash alpha\_m$.

This can also be written

:$\backslash Pr(\backslash mathbf\{x\}\backslash mid\; x\_0,\; \backslash alpha\_0,\; \backslash boldsymbol\{\backslash alpha\})=\backslash frac\{\backslash mathrm\{B\}(x\_\backslash bullet,\backslash alpha\_\backslash bullet)\}\{\backslash mathrm\{B\}(x\_0,\backslash alpha\_0)\}\backslash prod\_\{i=1\}^m\; \backslash frac\{\backslash Gamma(x\_i+\backslash alpha\_i)\}\{x\_i!\; \backslash Gamma(\backslash alpha\_i)\}.$

To obtain the marginal distribution over a subset of Dirichlet negative multinomial random variables, one only needs to drop the irrelevant $\backslash alpha\_i$'s (the variables that one wants to marginalize out) from the $\backslash boldsymbol\{\backslash alpha\}$ vector. The joint distribution of the remaining random variates is $\backslash mathrm\{DNM\}(x\_0,\backslash alpha\_0,\backslash boldsymbol\{\backslash alpha\_\{(-)\}\})$ where $\backslash boldsymbol\{\backslash alpha\_\{(-)\}\}$ is the vector with the removed $\backslash alpha\_i$'s. The univariate marginals are said to be beta negative binomially distributed.

If m-dimensional x is partitioned as follows

:$$

\mathbf{x}

=

\begin{bmatrix}

\mathbf{x}^{(1)} \\

\mathbf{x}^{(2)}

\end{bmatrix}

\text{ with sizes }\begin{bmatrix} q \times 1 \\ (m-q) \times 1 \end{bmatrix}

and accordingly $\backslash boldsymbol\{\backslash alpha\}$

:$$

\boldsymbol\alpha

=

\begin{bmatrix}

\boldsymbol\alpha^{(1)} \\

\boldsymbol\alpha^{(2)}

\end{bmatrix}

\text{ with sizes }\begin{bmatrix} q \times 1 \\ (m-q) \times 1 \end{bmatrix}

then the conditional distribution of $\backslash mathbf\{X\}^\{(1)\}$ on $\backslash mathbf\{X\}^\{(2)\}=\backslash mathbf\{x\}^\{(2)\}$ is $\backslash mathrm\{DNM\}(x\_0^\{\backslash prime\},\backslash alpha\_0^\{\backslash prime\},\backslash boldsymbol\backslash alpha^\{(1)\})$ where

:$$

x_0^{\prime} = x_0 + \sum_{i=1}^{m-q} x_i^{(2)}

and

:$$

\alpha_0^{\prime} = \alpha_0 + \sum_{i=1}^{m-q} \alpha_i^{(2)}

.

That is,

:$\backslash Pr(\backslash mathbf\{x\}^\{(1)\}\backslash mid\; \backslash mathbf\{x\}^\{(2)\},\; x\_0,\; \backslash alpha\_0,\; \backslash boldsymbol\{\backslash alpha\})=\; \backslash frac\{\backslash mathrm\{B\}(x\_\backslash bullet,\backslash alpha\_\backslash bullet)\}\{\backslash mathrm\{B\}(x\_0^\{\backslash prime\}\; ,\backslash alpha\_0^\{\backslash prime\})\; \}\backslash prod\_\{i=1\}^q\backslash frac\{\backslash Gamma(x\_i^\{(1)\}+\backslash alpha\_i^\{(1)\})\}\{(x\_i^\{(1)\}!)\backslash Gamma(\backslash alpha\_i^\{(1)\})\}$

The conditional distribution of a Dirichlet negative multinomial distribution on $\backslash sum\_\{i=1\}^m\; x\_i\; =\; n$ is Dirichlet-multinomial distribution with parameters $n$ and $\backslash boldsymbol\{\backslash alpha\}$. That is

:$\backslash Pr(\backslash mathbf\{x\}\; \backslash mid\; \backslash sum\_\{i=1\}^m\; x\_i\; =n,\; x\_0,\; \backslash alpha\_0,\; \backslash boldsymbol\{\backslash alpha\})=\; \backslash frac\{n!\backslash Gamma\backslash left(\backslash sum\_\{i=1\}^m\; \backslash alpha\_i\backslash right)\}$

{\Gamma\left(n+\sum_{i=1}^m \alpha_i\right)}\prod_{i=1}^m\frac{\Gamma(x_{i}+\alpha_{i})}{x_{i}!\Gamma(\alpha_{i})} .

Notice that the expression does not depend on $x\_0$ or $\backslash alpha\_0$.

If

:$X\; =\; (X\_1,\; \backslash ldots,\; X\_m)\backslash sim\backslash operatorname\{DNM\}(x\_0,\; \backslash alpha\_0,\; \backslash alpha\_1,\backslash ldots,\backslash alpha\_m)$

then, if the random variables with positive subscripts i and j are dropped from the vector and replaced by their sum,

:$X\text{'}\; =\; (X\_1,\; \backslash ldots,\; X\_i\; +\; X\_j,\; \backslash ldots,\; X\_m)\backslash sim\backslash operatorname\{DNM\}\; \backslash left(x\_0,\; \backslash alpha\_0,\; \backslash alpha\_1,\backslash ldots,\backslash alpha\_i+\backslash alpha\_j,\backslash ldots,\backslash alpha\_m\; \backslash right).$

For $\backslash alpha\_0>2$ the entries of the correlation matrix are

:$\backslash rho(X\_i,X\_i)\; =\; 1.$

:$\backslash rho(X\_i,X\_j)\; =\; \backslash frac\{\backslash operatorname\{cov\}(X\_i,X\_j)\}\{\backslash sqrt\{\backslash operatorname\{var\}(X\_i)\backslash operatorname\{var\}(X\_j)\}\}\; =\; \backslash sqrt\{\backslash frac\{\backslash alpha\_i\; \backslash alpha\_j\}\{(\backslash alpha\_0+\backslash alpha\_i-1)(\backslash alpha\_0+\backslash alpha\_j-1)\}\}.$

The Dirichlet negative multinomial is a heavy tailed distribution. It does not have a finite mean for $\backslash alpha\_0\; \backslash leq\; 1$ and it has infinite covariance matrix for $\backslash alpha\_0\; \backslash leq\; 2$. Therefore the moment generating function does not exist.

In the case when the $m+2$ parameters $x\_0,\; \backslash alpha\_0$ and $\backslash boldsymbol\{\backslash alpha\}$ are positive integers the Dirichlet negative multinomial can also be motivated by an urn model - or more specifically a basic Pólya urn model. Consider an urn initially containing $\backslash sum\_\{i=0\}^m\{\backslash alpha\_i\}$ balls of $m+1$ various colors including $\backslash alpha\_0$ red balls (the stopping color). The vector $\backslash boldsymbol\{\backslash alpha\}$ gives the respective counts of the other balls of various $m$ non-red colors. At each step of the model, a ball is drawn at random from the urn and replaced, along with one additional ball of the same color. The process is repeated over and over, until $x\_0$ red colored balls are drawn. The random vector $\backslash mathbf\{X\}$ of observed draws of the other $m$ non-red colors are distributed according to a $\backslash mathrm\{DNM\}(x\_0,\; \backslash alpha\_0,\; \backslash boldsymbol\{\backslash alpha\})$. Note, at the end of the experiment, the urn always contains the fixed number $x\_0+\backslash alpha\_0$ of red balls while containing the random number $\backslash mathbf\{X\}+\backslash boldsymbol\{\backslash alpha\}$ of the other $m$ colors.

• Beta negative binomial distribution
• Negative multinomial distribution
• Dirichlet-multinomial distribution

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Category:Multivariate discrete distributions