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Concept
Dirichlet negative multinomial distribution
Summary
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 multiple brands.
If parameters of the Dirichlet distribution are , and if
where
then the marginal distribution of X is a Dirichlet negative multinomial distribution:
In the above, is the negative multinomial distribution and is the Dirichlet 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 , 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:
which results in the following formula:
where and are the dimensional vectors created by appending the scalars and to the dimensional vectors and respectively and is the multivariate version of the beta function. We can write this equation explicitly as
Alternative formulations exist. One convenient representation is
where and .
This can also be written
To obtain the marginal distribution over a subset of Dirichlet negative multinomial random variables, one only needs to drop the irrelevant 's (the variables that one wants to marginalize out) from the vector. The joint distribution of the remaining random variates is where is the vector with the removed 's. The univariate marginals are said to be beta negative binomially distributed.
If m-dimensional x is partitioned as follows
and accordingly
then the conditional distribution of on is where
and
That is,
The conditional distribution of a Dirichlet negative multinomial distribution on is Dirichlet-multinomial distribution with parameters and .
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