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Dirichlet negative multinomial distribution
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<p>In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the Dirichlet negative multinomial distribution is a multivariate distribution on the non-negative integers. It is a multivariate extension of the <a href="/facts/Beta_negative_binomial_distribution/FdNmEqFQ">beta negative binomial distribution</a>. It is also a generalization of the <a href="/facts/Negative_multinomial_distribution/L5F1qvyl">negative multinomial distribution</a> (NM(<i>k</i>, <i>p</i>)) allowing for heterogeneity or <a href="/facts/Overdispersion/NSzgPcjq">overdispersion</a> to the probability vector. It is used in <a href="/facts/Quantitative_marketing_research/Gpd2vH14">quantitative marketing research</a> to flexibly model the number of household transactions across multiple brands. </p><p>If parameters of the <a href="/facts/Dirichlet_distribution/Qq13f5g2">Dirichlet distribution</a> are α {\displaystyle {\boldsymbol {\alpha }}} , and if </p> X ∣ p ∼ NM ( x 0 , p ) , {\displaystyle X\mid p\sim \operatorname {NM} (x_{0},\mathbf {p} ),} <p>where </p> p ∼ Dir ( α 0 , α ) , {\displaystyle \mathbf {p} \sim \operatorname {Dir} (\alpha _{0},{\boldsymbol {\alpha }}),} <p>then the marginal distribution of <i>X</i> is a Dirichlet negative multinomial distribution: </p> X ∼ DNM ( x 0 , α 0 , α ) . {\displaystyle X\sim \operatorname {DNM} (x_{0},\alpha _{0},{\boldsymbol {\alpha }}).} <p>In the above, NM ( x 0 , p ) {\displaystyle \operatorname {NM} (x_{0},\mathbf {p} )} is the <a href="/facts/Negative_multinomial_distribution/L5F1qvyl">negative multinomial distribution</a> and Dir ( α 0 , α ) {\displaystyle \operatorname {Dir} (\alpha _{0},{\boldsymbol {\alpha }})} is the <a href="/facts/Dirichlet_distribution/Qq13f5g2">Dirichlet distribution</a>. </p>