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In probability and statistics, the Dirichlet-multinomial distribution is a probability distribution for a multivariate discrete random variable. It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Pólya distribution (after George Pólya). It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector , and a set of discrete samples is drawn from the categorical distribution with probability vector p. The compounding corresponds to a Polya urn scheme. In document classification, for example, the distribution is used to represent the distributions of word counts for different document types. ==Probability mass function== Conceptually, we are doing ''N'' independent draws from a categorical distribution with ''K'' categories. Let us represent the independent draws as random categorical variables for . Let us denote the number of times a particular category has been seen (for ) among all the categorical variables as . Note that . Then, we have two separate views onto this problem: # A set of categorical variables . # A single vector-valued variable , distributed according to a multinomial distribution. The former case is a set of random variables specifying each ''individual'' outcome, while the latter is a variable specifying the ''number'' of outcomes of each of the ''K'' categories. The distinction is important, as the two cases have correspondingly different probability distributions. The parameter of the categorical distribution is where is the probability to draw value ; is likewise the parameter of the multinomial distribution . Rather than specifying directly, we give it a conjugate prior distribution, and hence it is drawn from a Dirichlet distribution with parameter vector . By integrating out , we obtain a compound distribution. However, the form of the distribution is different depending on which view we take. 抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)』 ■ウィキペディアで「Dirichlet-multinomial distribution」の詳細全文を読む スポンサード リンク
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