Multivariate Polya distribution

The multivariate Pólya distribution, also called the Dirichlet compound multinomial distribution, is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector α, and a set of discrete samples x is drawn from the multinomial distribution with probability vector p. The compounding corresponds to a Polya urn scheme. In document classification, for example, the distribution is used to represent probabilities over word counts for different document types.

The probability of a vector of counts x given the parameter vector α is obtained by integrating out the parameters p of the multinomial distribution:

which results in the following explicit formula:

where Γ is the gamma function, n_k is the number of times the outcome in x was k, and

is the total number of trials.

The two-dimensional version of the multivariate Pólya distribution is known as the Beta-binomial model.

The multivariate Pólya distribution is used in automated document classification and clustering, genetics, economy, combat modeling, and quantitative marketing.

See also

• Beta-binomial model
• Chinese restaurant process
• Dirichlet process
• Generalized Dirichlet distribution
• George Pólya
• Urn problem

References

• Elkan, C. (2006) Clustering documents with an exponential-family approximation of the Dirichlet compound multinomial distribution. ICML, 289-296
• Kvam, P. and Day, D. (2001) The multivariate Polya distribution in combat modeling. Naval Research Logistics, 48, 1-17
• Madsen, RE., Kauchak, D. and Elkan, C. (2005) Modeling Word Burstiness Using the Dirichlet Distribution. ICML, 545-552
• Minka, T. (2003) Estimating a Dirichlet distribution. Technical report Microsoft Research. Includes Matlab code for fitting distributions to data.
• Wagner, U. and Taudes, A. (1986) A Multivariate Polya Model of Brand Choice and Purchase Incidence. Marketing Science, 5(3), 219-244.