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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Dirichlet process ： ウィキペディア英語版 Dirichlet process In probability theory, Dirichlet processes (after Peter Gustav Lejeune Dirichlet) are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose domain is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables, that is, how likely it is that the random variables are distributed according to one or another particular distribution. The Dirichlet process is specified by a base distribution $H$ and a positive real number $\backslash alpha$ called the concentration parameter. The base distribution is the expected value of the process, that is, the Dirichlet process draws distributions "around" the base distribution in the way that a normal distribution draws real numbers around its mean. However, even if the base distribution is continuous, the distributions drawn from the Dirichlet process are almost surely discrete. The concentration parameter specifies how strong this discretization is: in the limit of $\backslash alpha\backslash rightarrow\; 0$, the realizations are all concentrated on a single value, while in the limit of $\backslash alpha\backslash rightarrow\backslash infty$ the realizations become continuous. In between the two extremes the realizations are discrete distributions with less and less concentration as $\backslash alpha$ increases. The Dirichlet process can also be seen as the infinite-dimensional generalization of the Dirichlet distribution. In the same way as the Dirichlet distribution is the conjugate prior for the categorical distribution, the Dirichlet process is the conjugate prior for infinite, nonparametric discrete distributions. A particularly important application of Dirichlet processes is as a prior probability distribution in infinite mixture models. The Dirichlet process was formally introduced by Thomas Ferguson in 1973 and has since been applied in data mining and machine learning, among others for natural language processing, computer vision and bioinformatics. ==Introduction== Dirichlet processes are usually used when modeling data that tends to repeat previous values in a "rich get richer" fashion. Specifically, suppose that the generation of values $X\_,X\_,\backslash dots$ can be simulated by the following algorithm. :Input: $H$ (a probability distribution called base distribution), $\backslash alpha$ (a positive real number called concentration parameter) # Draw $X\_$ from the distribution $H$. # For $n>1$: ## With probability $\backslash frac$ draw $X\_$ from $H$. ## With probability $\backslash frac$ set $X\_=x$, where $n\_$ is the number of previous observations At the same time, another common model for data is that the observations $X\_,X\_,\backslash dots$ are assumed to be independent and identically distributed (i.i.d.) according to some distribution $P$. The goal in introducing Dirichlet processes is to be able to describe the procedure outlined above in this i.i.d. model. The $X\_,X\_,\backslash dots$ observations are not independent, since we have to consider the previous results when generating the next value. They are, however, exchangeable. This fact can be shown by calculating the joint probability distribution of the observations and noticing that the resulting formula only depends on which $x$ values occur among the observations and how many repetitions they each have. Because of this exchangeability, de Finetti's representation theorem applies and it implies that the observations $X\_,X\_,\backslash dots$ are conditionally independent given a (latent) distribution $P$. This $P$ is a random variable itself and has a distribution. This distribution (over distributions) is called Dirichlet process ($\backslash mathrm$). In summary, this means that we get an equivalent procedure to the above algorithm: # Draw a distribution $P$ from $\backslash mathrm\backslash left(H,\backslash alpha\backslash right)$ # Draw observations $X\_,X\_\backslash dots$ independently from $P$. In practice, however, drawing a concrete distribution $P$ is impossible, since its specification requires an infinite amount of information. This is a common phenomenon in the context of Bayesian non-parametric statistics where a typical task is to learn distributions on function spaces, which involve effectively infinitely many parameters. The key insight is that in many applications the infinite dimensional distributions appear only as an intermediary computational device and are not required for either the initial specification of prior beliefs or for the statement of the final inference. The Dirichlet process can be used to circumvent infinite computational requirements as described above. 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 process」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.