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Empirical Bayes methods are procedures for statistical inference in which the prior probability 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.
The equation of the OLS estimator in a hypothetical regression of the population means on the sample means gives an estimator of the form of either the James–Stein estimator (when we force the OLS intercept to equal 0) or of the Efron-Morris estimator (when we allow the intercept to vary).
A Bayes estimator derived through the empirical Bayes method is called an empirical Bayes estimator. Empirical Bayes methods enable the use of auxiliary empirical data, from observations of related parameters, in the development of a Bayes estimator. This is done under the assumption that the estimated parameters are obtained from a common prior.
But the adjustment formula yields an artificial shrinkage. A shrinkage estimator is an estimator that, either explicitly or implicitly, incorporates the effects of shrinkage. In loose terms this means that a naive or raw estimate is improved by combining it with other information.
Best linear unbiased predictions" (BLUPs) of random effects are similar to best linear unbiased estimates (BLUEs) (see Gauss–Markov theorem) of fixed effects. The distinction arises because it is conventional to talk about estimating fixed effects but about predicting random effects, but the two terms are otherwise equivalent. (This is a bit ...
The objective of Bayesian model reduction is to compute the posterior ~ and evidence ~ of the reduced model from the posterior () and evidence () of the full model. . Combining Equation 1 and Equation 2 and re-arranging, the reduced posterior ~ can be expressed as the product of the full posterior, the ratio of priors and the ratio of evi
For large samples, the shrinkage intensity will reduce to zero, hence in this case the shrinkage estimator will be identical to the empirical estimator. Apart from increased efficiency the shrinkage estimate has the additional advantage that it is always positive definite and well conditioned. Various shrinkage targets have been proposed:
It can be drastically simplified by assuming that the probability of appearance of a word knowing the nature of the text (spam or not) is independent of the appearance of the other words. This is the naive Bayes assumption and this makes this spam filter a naive Bayes model. For instance, the programmer can assume that: