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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 Imprecise Dirichlet process ： ウィキペディア英語版 Imprecise Dirichlet process In probability theory and statistics, the Dirichlet process (DP) is one of the most popular Bayesian nonparametric models. It was introduced by Ferguson as a prior over probability distributions. A Dirichlet process $\backslash mathrm\backslash left(s,G\_0\backslash right)$ is completely defined by its parameters: $G\_0$ (the ''base distribution'' or ''base measure'') is an arbitrary distribution and $s$ (the ''concentration parameter'') is a positive real number (it is often denoted as $\backslash alpha$). According to the Bayesian paradigma these parameters should be chosen based on the available prior information on the domain. The question is: how should we choose the prior parameters $\backslash left(s,G\_0\backslash right)$ of the DP, in particular the infinite dimensional one $G\_0$, in case of lack of prior information? To address this issue, the only prior that has been proposed so far is the limiting DP obtained for $s\backslash rightarrow\; 0$, which has been introduced under the name of Bayesian bootstrap by Rubin;〔Rubin D (1981). The Bayesian bootstrap. Ann Statist 9 130–134〕 in fact it can be proven that the Bayesian bootstrap is asymptotically equivalent to the frequentist bootstrap introduced by Bradley Efron.〔Efron B (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26〕 The limiting Dirichlet process $s\backslash rightarrow\; 0$ has been criticized on diverse grounds. From an a-priori point of view, the main criticism is that taking $s\backslash rightarrow\; 0$ is far from leading to a noninformative prior. Moreover, a-posteriori, it assigns zero probability to any set that does not include the observations.〔Rubin D (1981). The Bayesian bootstrap. Ann Statist 9 130–134〕 The imprecise Dirichlet process has been proposed to overcome these issues. The basic idea is to fix $s\; >\; 0$ but do not choose any precise base measure $G\_0$. More precisely, the imprecise Dirichlet process (IDP) is defined as follows: : $$ ~~\mathrm:~\left\\right\} where $\backslash mathbb$ is the set of all probability measures. In other words, the IDP is the set of all Dirichlet processes (with a fixed $s\; >\; 0$) obtained by letting the base measure $G\_0$ to span the set of all probability measures. == Inferences with the Imprecise Dirichlet Process == Let $P$ a probability distribution on $(\backslash mathbb,\backslash mathcal)$ (here $\backslash mathbb$ is a standard Borel space with Borel $\backslash sigma$-field $\backslash mathcal$) and assume that $P\backslash sim\; \backslash mathrm(s,G\_0)$. Then consider a real-valued bounded function $f$ defined on $(\backslash mathbb,\backslash mathcal)$. It is well known that the expectation of $E()$ with respect to the Dirichlet process is : $$ \mathcal (E(f) )=\mathcal\left(f \, dP\right )=\int f \,d\mathcal() = \int f \, dG_0. One of the most remarkable properties of the DP priors is that the posterior distribution of $P$ is again a DP. Let $X\_1,\backslash dots,X\_n$ be an independent and identically distributed sample from $P$ and $P\; \backslash sim\; Dp(s,G\_0)$, then the posterior distribution of $P$ given the observations is : $$ P\mid X_1,\dots,X_n \sim Dp\left(s+n, G_n\right),~~~ \text~~~~~~ G_n=\frac G_0+ \frac \sum\limits_^n \delta_, where $\backslash delta\_$ is an atomic probability measure (Dirac's delta) centered at $X\_i$. Hence, it follows that $\backslash mathcal(X\_1,\backslash dots,X\_n)=\; \backslash int\; f\; \backslash ,\; dG\_n.$ Therefore, for any fixed $G\_0$, we can exploit the previous equations to derive prior and posterior expectations. In the IDP $G\_0$ can span the set of all distributions $\backslash mathbb$. This implies that we will get a different prior and posterior expectation of $E(f)$ for any choice of $G\_0$. A way to characterize inferences for the IDP is by computing lower and upper bounds for the expectation of $E(f)$ w.r.t. $G\_0\; \backslash in\; \backslash mathbb$. A-priori these bounds are: :$$ \underline} \int f \,dG_0=\inf f, ~~~~\overline} \int f \,dG_0=\sup f, the lower (upper) bound is obtained by a probability measure that puts all the mass on the infimum (supremum) of $f$, i.e., $G\_0=\backslash delta\_$ with $X\_0=\backslash arg\; \backslash inf\; f$ (or respectively with $X\_0=\backslash arg\; \backslash sup\; f$). From the above expressions of the lower and upper bounds, it can be observed that the range of $\backslash mathcal()$ under the IDP is the same as the original range of $f$. In other words, by specifying the IDP, we are not giving any prior information on the value of the expectation of $f$. A-priori, IDP is therefore a model of prior (near)-ignorance for $E(f)$. A-posteriori, IDP can learn from data. The posterior lower and upper bounds for the expectation of $E(f)$ are in fact given by: :$$ \begin \underline} \int f \, dG_n = \frac \inf f+ \int f(X) \frac \sum\limits_^n \delta_(dX) \\ & =\frac \inf f+ \frac \frac,\\() \overline} \int f \, dG_n= \frac \sup f+ \int f(X) \frac \sum\limits_^n \delta_(dX) \\ & =\frac \sup f+ \frac \frac. \end It can be observed that the posterior inferences do not depend on $G\_0$. To define the IDP, the modeler has only to choose $s$ (the concentration parameter). This explains the meaning of the adjective ''near'' in prior near-ignorance, because the IDP requires by the modeller the elicitation of a parameter. However, this is a simple elicitation problem for a nonparametric prior, since we only have to choose the value of a positive scalar (there are not infinitely many parameters left in the IDP model). Finally, observe that for $n\; \backslash rightarrow\; \backslash infty$, IDP satisfies :$$ \underline} \left(\mid X_1,\dots,X_n\right ) \rightarrow S(f), where $S(f)=\backslash lim\_\; \backslash tfrac\backslash sum\_^n\; f(X\_i)$. In other words, the IDP is consistent. 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「Imprecise 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.