Dirichlet negative multinomial distribution

In probability theory and statistics, the Dirichlet negative multinomial distribution is a multivariate distribution on the non-negative integers. It is a multivariate extension of the beta negative binomial distribution. It is also a generalization of the negative multinomial distribution (NM(k, p)) allowing for heterogeneity or overdispersion to the probability vector. It is used in quantitative marketing research to flexibly model the number of household transactions across several brands.

Notation	${\displaystyle {\textrm {DNM}}(x_{0},\,\alpha _{0},\,{\boldsymbol {\alpha }})}$
Parameters	${\displaystyle x_{0}\in R,\alpha _{0}\in R,{\boldsymbol {\alpha }}\in R^{m}}$
Support	${\displaystyle x_{i}\in \{0,1,2,\ldots \},1\leq i\leq m}$
PDF	${\displaystyle {\frac {\mathrm {B} (\sum _{i=0}^{m}x_{i},\sum _{i=0}^{m}\alpha _{i})}{\mathrm {B} (x_{0},\alpha _{0})}}\prod _{i=1}^{m}{\frac {\Gamma (x_{i}+\alpha _{i})}{x_{i}!\Gamma (\alpha _{i})}}}$ where Γ(x) is the Gamma function and B is the beta function.
Mean	${\displaystyle {\tfrac {x_{0}}{\alpha _{0}-1}}{\boldsymbol {\alpha }}}$ for ${\displaystyle \alpha _{0}>1}$
Variance	${\displaystyle \,{\frac {x_{0}(x_{0}+\alpha _{0}-1)}{(\alpha _{0}-1)^{2}(\alpha _{0}-2)}}\left[{\boldsymbol {\alpha }}{\boldsymbol {\alpha }}'+(\alpha _{0}-1)\operatorname {diag} ({\boldsymbol {\alpha }})\right]}$ for ${\displaystyle \alpha _{0}>2}$
MGF	undefined

If parameters of the Dirichlet distribution are ${\displaystyle {\boldsymbol {\alpha }}}$, and if

${\displaystyle X\mid p\sim \operatorname {NM} (x_{0},\mathbf {p} ),}$

where

${\displaystyle \mathbf {p} \sim \operatorname {Dir} (\alpha _{0},{\boldsymbol {\alpha }}),}$

then the marginal distribution of X is a Dirichlet negative multinomial distribution:

${\displaystyle X\sim \operatorname {DNM} (x_{0},\alpha _{0},{\boldsymbol {\alpha }}).}$

In the above, ${\displaystyle \operatorname {NM} (x_{0},\mathbf {p} )}$ is the negative multinomial distribution and ${\displaystyle \operatorname {Dir} (\alpha _{0},{\boldsymbol {\alpha }})}$ is the Dirichlet distribution.

Motivation

Dirichlet negative multinomial as a compound 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 ${\displaystyle \mathbf {x} =(x_{1},\dots ,x_{K})}$, 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:

${\displaystyle \Pr(\mathbf {x} \mid x_{0},\alpha _{0},{\boldsymbol {\alpha }})=\int _{\mathbf {p} }\Pr(\mathbf {x} \mid x_{0},\mathbf {p} )\Pr(\mathbf {p} \mid \alpha _{0},{\boldsymbol {\alpha }}){\textrm {d}}\mathbf {p} }$

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

which results in the following formula:

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

where ${\displaystyle \mathbf {x_{+}} }$ and ${\displaystyle {\boldsymbol {\alpha }}_{+}}$ are the ${\displaystyle m+1}$ dimensional vectors created by appending the scalars ${\displaystyle x_{0}}$ and ${\displaystyle \alpha _{0}}$ to the ${\displaystyle m}$ dimensional vectors ${\displaystyle \mathbf {x} }$ and ${\displaystyle {\boldsymbol {\alpha }}}$ respectively and ${\displaystyle \mathrm {B} }$ is the multivariate version of the beta function. We can write this equation explicitly as

${\displaystyle \Pr(\mathbf {x} \mid x_{0},\alpha _{0},{\boldsymbol {\alpha }})=x_{0}{\frac {\Gamma (\sum _{i=0}^{m}x_{i})\Gamma (\sum _{i=0}^{m}\alpha _{i})}{\Gamma (\sum _{i=0}^{m}(x_{i}+\alpha _{i}))}}\prod _{i=0}^{m}{\frac {\Gamma (x_{i}+\alpha _{i})}{\Gamma (x_{i}+1)\Gamma (\alpha _{i})}}.}$

Alternative formulations exist. One convenient representation[1] is

${\displaystyle \Pr(\mathbf {x} \mid x_{0},\alpha _{0},{\boldsymbol {\alpha }})={\frac {\Gamma (x_{\bullet })}{\Gamma (x_{0})\prod _{i=1}^{m}\Gamma (x_{i}+1)}}\times {\frac {\Gamma (\alpha _{\bullet })}{\prod _{i=0}^{m}\Gamma (\alpha _{i})}}\times {\frac {\prod _{i=0}^{m}\Gamma (x_{i}+\alpha _{i})}{\Gamma (x_{\bullet }+\alpha _{\bullet })}}}$

where ${\displaystyle x_{\bullet }=x_{0}+x_{1}+\cdots +x_{m}}$ and ${\displaystyle \alpha _{\bullet }=\alpha _{0}+\alpha _{1}+\cdots +\alpha _{m}}$.

This can also be written

${\displaystyle \Pr(\mathbf {x} \mid x_{0},\alpha _{0},{\boldsymbol {\alpha }})={\frac {\mathrm {B} (x_{\bullet },\alpha _{\bullet })}{\mathrm {B} (x_{0},\alpha _{0})}}\prod _{i=1}^{m}{\frac {\Gamma (x_{i}+\alpha _{i})}{x_{i}!\Gamma (\alpha _{i})}}.}$

Properties

Marginal distributions

To obtain the marginal distribution over a subset of Dirichlet negative multinomial random variables, one only needs to drop the irrelevant ${\displaystyle \alpha _{i}}$'s (the variables that one wants to marginalize out) from the ${\displaystyle {\boldsymbol {\alpha }}}$ vector. The joint distribution of the remaining random variates is ${\displaystyle \mathrm {DNM} (x_{0},\alpha _{0},{\boldsymbol {\alpha _{-}}})}$ where ${\displaystyle {\boldsymbol {\alpha _{-}}}}$ is the vector with the removed ${\displaystyle \alpha _{i}}$'s.

Conditional distributions

If m-dimensional x is partitioned as follows

${\displaystyle \mathbf {x} ={\begin{bmatrix}\mathbf {x} ^{(1)}\\\mathbf {x} ^{(2)}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(m-q)\times 1\end{bmatrix}}}$

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

${\displaystyle {\boldsymbol {\alpha }}={\begin{bmatrix}{\boldsymbol {\alpha }}^{(1)}\\{\boldsymbol {\alpha }}^{(2)}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(m-q)\times 1\end{bmatrix}}}$

then the conditional distribution of ${\displaystyle \mathbf {x} ^{(1)}}$ on ${\displaystyle \mathbf {x} ^{(2)}}$ is ${\displaystyle \mathrm {DNM} (x_{0}^{\prime },\alpha _{0}^{\prime },{\boldsymbol {\alpha }}^{(1)})}$ where

${\displaystyle x_{0}^{\prime }=x_{0}+\sum _{i=1}^{m-q}x_{i}^{(2)}}$

and

${\displaystyle \alpha _{0}^{\prime }=\alpha _{0}+\sum _{i=1}^{m-q}\alpha _{i}^{(2)}}$.

That is,

${\displaystyle \Pr(\mathbf {x} ^{(1)}\mid \mathbf {x} ^{(2)},x_{0},\alpha _{0},{\boldsymbol {\alpha }})={\frac {\mathrm {B} (x_{\bullet },\alpha _{\bullet })}{\mathrm {B} (x_{0}^{\prime },\alpha _{0}^{\prime })}}\prod _{i=1}^{q}{\frac {\Gamma (x_{i}^{(1)}+\alpha _{i}^{(1)})}{(x_{i}^{(1)}!)\Gamma (\alpha _{i}^{(1)})}}}$

Conditional on the sum

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

${\displaystyle \Pr(\mathbf {x} \mid \sum _{i=1}^{m}x_{i}=n,x_{0},\alpha _{0},{\boldsymbol {\alpha }})={\frac {n!\Gamma \left(\sum _{i=1}^{m}\alpha _{i}\right)}{\Gamma \left(n+\sum _{i=1}^{m}\alpha _{i}\right)}}\prod _{i=1}^{m}{\frac {\Gamma (x_{i}+\alpha _{i})}{x_{i}!\Gamma (\alpha _{i})}}}$.

Notice that the equation does not depend on ${\displaystyle x_{0}}$ or ${\displaystyle \alpha _{0}}$.

Correlation matrix

For ${\displaystyle \alpha _{0}>2}$ the entries of the correlation matrix are

${\displaystyle \rho (X_{i},X_{i})=1.}$

${\displaystyle \rho (X_{i},X_{j})={\frac {\operatorname {cov} (X_{i},X_{j})}{\sqrt {\operatorname {var} (X_{i})\operatorname {var} (X_{j})}}}={\sqrt {\frac {\alpha _{i}\alpha _{j}}{(\alpha _{0}+\alpha _{i}-1)(\alpha _{0}+\alpha _{j}-1)}}}.}$

Heavy tailed

The Dirichlet negative multinomial is a heavy tailed distribution. It does not have a finite mean for ${\displaystyle \alpha _{0}\leq 1}$ and it has infinite covariance matrix for ${\displaystyle \alpha _{0}\leq 2}$. It therefore has undefined moment generating function.

Aggregation

If

${\displaystyle X=(X_{1},\ldots ,X_{m})\sim \operatorname {DNM} (x_{0},\alpha _{0},\alpha _{1},\ldots ,\alpha _{m})}$

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

${\displaystyle X'=(X_{1},\ldots ,X_{i}+X_{j},\ldots ,X_{m})\sim \operatorname {DNM} \left(x_{0},\alpha _{0},\alpha _{1},\ldots ,\alpha _{i}+\alpha _{j},\ldots ,\alpha _{m}\right).}$

Applications

Dirichlet negative multinomial as a urn model

The Dirichlet negative multinomial can also be motivated by an urn model in the case when ${\displaystyle x_{0}}$ is a positive integer. Consider a sequence of independent and identically distributed multinomial trials, each of which has ${\displaystyle 1+m}$ outcomes. Call one of the outcomes a “success”, and suppose it has probability ${\displaystyle p_{0}}$. The other ${\displaystyle m}$ outcomes – called "failures" - have probabilities ${\displaystyle p_{1},p_{2},\ldots ,p_{m}}$. If the vector ${\displaystyle (x_{1},x_{2},\cdots x_{m})}$ counts the m types of failures before the ${\displaystyle x_{0}}$ success is observed, then the ${\displaystyle (x_{1},x_{2},\ldots ,x_{m})}$ have negative mulitnomial distribution with parameters ${\displaystyle (x_{0},p_{1},\ldots ,p_{m})}$.

If the parameters ${\displaystyle (p_{0},p_{1},\ldots ,p_{m})}$ are themselves sampled from a Dirichlet distribution with parameters ${\displaystyle (\alpha _{0},\alpha _{1},\ldots ,\alpha _{m})}$, then the resulting distribution of ${\displaystyle (x_{1},x_{2},\ldots ,x_{m})}$ is Dirichlet negative multinomial. The resultant distribution has ${\displaystyle 2+m}$ parameters.

See also

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

References

1. Farewell, Daniel & Farewell, Vernon. (2012). Dirichlet negative multinomial regression for overdispersed correlated count data. Biostatistics (Oxford, England). 14. 10.1093/biostatistics/kxs050.