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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 Conjugate prior ： ウィキペディア英語版 Conjugate prior In Bayesian probability theory, if the posterior distributions ''p''(θ|''x'') are in the same family as the prior probability distribution ''p''(θ), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function. For example, the Gaussian family is conjugate to itself (or ''self-conjugate'') with respect to a Gaussian likelihood function: if the likelihood function is Gaussian, choosing a Gaussian prior over the mean will ensure that the posterior distribution is also Gaussian. This means that the Gaussian distribution is a conjugate prior for the likelihood that is also Gaussian. The concept, as well as the term "conjugate prior", were introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory.〔Howard Raiffa and Robert Schlaifer. ''Applied Statistical Decision Theory''. Division of Research, Graduate School of Business Administration, Harvard University, 1961.〕 A similar concept had been discovered independently by George Alfred Barnard.〔Jeff Miller et al. (Earliest Known Uses of Some of the Words of Mathematics ), ("conjugate prior distributions" ). Electronic document, revision of November 13, 2005, retrieved December 2, 2005.〕 Consider the general problem of inferring a distribution for a parameter θ given some datum or data ''x''. From Bayes' theorem, the posterior distribution is equal to the product of the likelihood function $\backslash theta\; \backslash mapsto\; p(x\backslash mid\backslash theta)\backslash !$ and prior $p(\; \backslash theta\; )\backslash !$, normalized (divided) by the probability of the data $p(\; x\; )\backslash !$: :$p(\backslash theta|x)\; =\; \backslash frac$ . \! Let the likelihood function be considered fixed; the likelihood function is usually well-determined from a statement of the data-generating process. It is clear that different choices of the prior distribution ''p''(θ) may make the integral more or less difficult to calculate, and the product ''p''(''x''|θ) × ''p''(θ) may take one algebraic form or another. For certain choices of the prior, the posterior has the same algebraic form as the prior (generally with different parameter values). Such a choice is a ''conjugate prior''. A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise a difficult numerical integration may be necessary. Further, conjugate priors may give intuition, by more transparently showing how a likelihood function updates a prior distribution. All members of the exponential family have conjugate priors.〔For a catalog, see 〕 == Example == The form of the conjugate prior can generally be determined by inspection of the probability density or probability mass function of a distribution. For example, consider a random variable which consists of the number of successes in ''n'' Bernoulli trials with unknown probability of success ''q'' in (). This random variable will follow the binomial distribution, with a probability mass function of the form :$p(x)\; =\; q^x\; (1-q)^$ Expressed as a function of $q$, this has the form :$f(q)\; \backslash propto\; q^a\; (1-q)^b$ for some constants $a$ and $b$. Generally, this functional form will have an additional multiplicative factor (the normalizing constant) ensuring that the function is a probability distribution, i.e. the integral over the entire range is 1. This factor will often be a function of $a$ and $b$, but never of $q$. In fact, the usual conjugate prior is the beta distribution with parameters ($\backslash alpha$, $\backslash beta$): :$p(q)\; =\; \backslash over\; \backslash Beta(\backslash alpha,\backslash beta)\}$ where $\backslash alpha$ and $\backslash beta$ are chosen to reflect any existing belief or information ($\backslash alpha$ = 1 and $\backslash beta$ = 1 would give a uniform distribution) and ''Β''($\backslash alpha$, $\backslash beta$) is the Beta function acting as a normalising constant. In this context, $\backslash alpha$ and $\backslash beta$ are called ''hyperparameters'' (parameters of the prior), to distinguish them from parameters of the underlying model (here ''q''). It is a typical characteristic of conjugate priors that the dimensionality of the hyperparameters is one greater than that of the parameters of the original distribution. If all parameters are scalar values, then this means that there will be one more hyperparameter than parameter; but this also applies to vector-valued and matrix-valued parameters. (See the general article on the exponential family, and consider also the Wishart distribution, conjugate prior of the covariance matrix of a multivariate normal distribution, for an example where a large dimensionality is involved.) If we then sample this random variable and get ''s'' successes and ''f'' failures, we have :$P(s,f|q=x)\; =\; x^s(1-x)^f,$ : & = (1-x)^ / \Beta(\alpha,\beta)} \over \int_^1 \left( y^(1-y)^ / \Beta(\alpha,\beta)\right) dy} \\ & = \over \Beta(s+\alpha,f+\beta)}, \\ \end which is another Beta distribution with parameters ($\backslash alpha$ + ''s'', $\backslash beta$ + ''f''). This posterior distribution could then be used as the prior for more samples, with the hyperparameters simply adding each extra piece of information as it comes. 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「Conjugate prior」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.