Dirichlet Multinomial Distribution Model
Dirichlet multinomial distribution model is a term used in statistics and probability and refers to the probability distribution for multivariate random, discrete variable. It is also known as multivariate Polya distribution or Dirichlet compound multinomial distribution (DCM).
It is a probability compound distribution in which the p vector probability is drawn from Dirichlet distribution that has a parameter vector as well as discrete samples that are drawn from categorical distribution with the vector p probability.
Dirichlet multinomial distribution model is often used as multinomial variables or prior categorical variables distribution in mixture models of Bayesian and other hierarchical Bayesian models. It is essential to note that in most fields like in natural language processing, there are categorical variables which are often imprecisely known as ‘multinomial variables’.
It is likely that such use can at times lead to confusion just as if binomial and Bernoulli distributions were conflated. Inference is often done over hierarchical Bayesian models through use of Gibbs sampling and in such instances of this distribution model are marginalized out of integration of the random variable. This in turn leads to categorical variables that are drawn from similar Dirichlet distribution.
In the large Bayesian network, distributions will occur with the distribution mode priors as part of a much larger network. Dirichlet priors can all be collapsed as long as the nodes that depend on them are categorical distributions. The collapse happens for every Dirichlet distribution node in a separate fashion from the others. This will occur regardless of whether there are any other nodes that depend on categorical distributions. It will also occur regardless of categorical distributions that depend in the nodes to Dirichlet priors. Ideally, all categorical distributions that are dependent on the given Dirichlet distribution nodes become connected to a single multinomial Dirichlet joint distribution.
There are instances when it is possible to have a hierarchical model in which there are multiple Dirichlet priors as a set of dependent variables and as before. Often, this presents a tricky situation though the relationship been the dependent variables and the priors is not fixed. Rather, the choice on the prior to be used is dependent on a different categorical variable.
This will occur in cases of topic models and names of variables that are above what is meant to match with the latent Dirichlet allocation. In such cases, the set W becomes a set of words with each drawn from a K of possible topics where every topic is a Dirichlet prior of V possible words that specify frequency of varying words of the topic.
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