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(()) = p_i (1-p_i) | char = where | pgf = | conjugate =Dirichlet: | }} In probability theory and statistics, a categorical distribution (also called a "generalized Bernoulli distribution", "multinoulli distribution" 〔Murphy, K. P. (2012). ''Machine learning: a probabilistic perspective'', p. 35. MIT press. ISBN 0262018020.〕 or, less precisely, a "discrete distribution") is a probability distribution that describes the possible results of a random event that can take on one of ''K'' possible outcomes, with the probability of each outcome separately specified. There is not necessarily an underlying ordering of these outcomes, but numerical labels are often attached for convenience in describing the distribution, (e.g. 1 to ''K''). Note that the ''K''-dimensional categorical distribution is the most general distribution over a ''K''-way event; any other discrete distribution over a size-''K'' sample space is a special case. The parameters specifying the probabilities of each possible outcome are constrained only by the fact that each must be in the range 0 to 1, and all must sum to 1. The categorical distribution is the generalization of the Bernoulli distribution for a categorical random variable, i.e. for a discrete variable with more than two possible outcomes. ==Terminology== Occasionally, the categorical distribution is termed the "discrete distribution". However, this properly refers not to one particular family of distributions but to a general class of distributions. Note that, in some fields, such as machine learning and natural language processing, the categorical and multinomial distributions are conflated, and it is common to speak of a "multinomial distribution" when a categorical distribution is actually meant.〔Minka, T. (2003) (Bayesian inference, entropy and the multinomial distribution ). Technical report Microsoft Research.〕 This imprecise usage stems from the fact that it is sometimes convenient to express the outcome of a categorical distribution as a "1-of-K" vector (a vector with one element containing a 1 and all other elements containing a 0) rather than as an integer in the range 1 to ''K''; in this form, a categorical distribution is equivalent to a multinomial distribution for a single observation (see below). However, conflating the categorical and multinomial distributions can lead to problems. For example, in a Dirichlet-multinomial distribution, which arises commonly in natural language processing models (although not usually with this name) as a result of collapsed Gibbs sampling where Dirichlet distributions are collapsed out of a Hierarchical Bayesian model, it is very important to distinguish categorical from multinomial. The joint distribution of the same variables with the same Dirichlet-multinomial distribution has two different forms depending on whether it is characterized as a distribution whose domain is over individual categorical nodes or over multinomial-style counts of nodes in each particular category (similar to the distinction between a set of Bernoulli-distributed nodes and a single binomial-distributed node). Both forms have very similar-looking probability mass functions (PMF's), which both make reference to multinomial-style counts of nodes in a category. However, the multinomial-style PMF has an extra factor, a multinomial coefficient, that is a constant equal to 1 in the categorical-style PMF. Confusing the two can easily lead to incorrect results in settings where this extra factor is not constant with respect to the distributions of interest. The factor is frequently constant in the complete conditionals used in Gibbs sampling and the optimal distributions in variational methods. 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「Categorical distribution」の詳細全文を読む スポンサード リンク
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