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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 Empirical Bayes method ： ウィキペディア英語版 Empirical Bayes method Empirical Bayes methods are procedures for statistical inference in which the prior distribution is estimated from the data. This approach stands in contrast to standard Bayesian methods, for which the prior distribution is fixed before any data are observed. Despite this difference in perspective, empirical Bayes may be viewed as an approximation to a fully Bayesian treatment of a hierarchical model wherein the parameters at the highest level of the hierarchy are set to their most likely values, instead of being integrated out. Empirical Bayes, also known as maximum marginal likelihood,〔 represents one approach for setting hyperparameters. ==Introduction== Empirical Bayes methods can be seen as an approximation to a fully Bayesian treatment of a hierarchical Bayes model. In, for example, a two-stage hierarchical Bayes model, observed data $y\; =\; \backslash$ are assumed to be generated from an unobserved set of parameters $\backslash theta\; =\; \backslash$ according to a probability distribution $p(y|\backslash theta)\backslash ,$. In turn, the parameters $\backslash theta$ can be considered samples drawn from a population characterised by hyperparameters $\backslash eta\backslash ,$ according to a probability distribution $p(\backslash theta|\backslash eta)\backslash ,$. In the hierarchical Bayes model, though not in the empirical Bayes approximation, the hyperparameters $\backslash eta\backslash ,$ are considered to be drawn from an unparameterized distribution $p(\backslash eta)\backslash ,$. Information about a particular quantity of interest $\backslash theta\_i\backslash ;$ therefore comes not only from the properties of those data which directly depend on it, but also from the properties of the population of parameters $\backslash theta\backslash ;$ as a whole, inferred from the data as a whole, summarised by the hyperparameters $\backslash eta\backslash ;$. Using Bayes' theorem, :$$ p(\theta|y) = \frac = \frac \int p(\theta | \eta) p(\eta) \, d\eta \,. In general, this integral will not be tractable analytically or symbolically and must be evaluated by numerical methods. Stochastic (random) or deterministic approximations may be used. Example stochastic methods are Markov Chain Monte Carlo and Monte Carlo sampling. Deterministic approximations are discussed in quadrature. Alternatively, the expression can be written as :$$ p(\theta|y) = \int p(\theta|\eta, y) p(\eta | y) \; d \eta = \int \frac p(\eta | y) \; d \eta\,, and the term in the integral can in turn be expressed as :$$ p(\eta | y) = \int p(\eta | \theta) p(\theta | y) \; d \theta . These suggest an iterative scheme, qualitatively similar in structure to a Gibbs sampler, to evolve successively improved approximations to $p(\backslash theta|y)\backslash ;$ and $p(\backslash eta|y)\backslash ;$. First, calculate an initial approximation to $p(\backslash theta|y)\backslash ;$ ignoring the $\backslash eta$ dependence completely; then calculate an approximation to $p(\backslash eta|y)\backslash ;$ based upon the initial approximate distribution of $p(\backslash theta|y)\backslash ;$; then use this $p(\backslash eta|y)\backslash ;$ to update the approximation for $p(\backslash theta|y)\backslash ;$; then update $p(\backslash eta|y)\backslash ;$; and so on. When the true distribution $p(\backslash eta|y)\backslash ;$ is sharply peaked, the integral determining $p(\backslash theta|y)\backslash ;$ may be not much changed by replacing the probability distribution over $\backslash eta\backslash ;$ with a point estimate $\backslash eta^\backslash ;$ representing the distribution's peak (or, alternatively, its mean), :$$ p(\theta|y) \simeq \frac\,. With this approximation, the above iterative scheme becomes the EM algorithm. The term "Empirical Bayes" can cover a wide variety of methods, but most can be regarded as an early truncation of either the above scheme or something quite like it. Point estimates, rather than the whole distribution, are typically used for the parameter(s) $\backslash eta\backslash ;$. The estimates for $\backslash eta^\backslash ;$ are typically made from the first approximation to $p(\backslash theta|y)\backslash ;$ without subsequent refinement. These estimates for $\backslash eta^\backslash ;$ are usually made without considering an appropriate prior distribution for $\backslash eta$. 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「Empirical Bayes method」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.