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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 = {,, …,} are assumed to be generated from an unobserved set of parameters = {,, …,} according to a probability distribution ().
The term relates to the notion that the improved estimate is made closer to the value supplied by the 'other information' than the raw estimate. In this sense, shrinkage is used to regularize ill-posed inference problems. Shrinkage is implicit in Bayesian inference and penalized likelihood inference, and explicit in James–Stein-type
The James–Stein estimator may seem at first sight to be a result of some peculiarity of the problem setting. In fact, the estimator exemplifies a very wide-ranging effect; namely, the fact that the "ordinary" or least squares estimator is often inadmissible for simultaneous estimation of several parameters.
All of these approaches rely on the concept of shrinkage. This is implicit in Bayesian methods and in penalized maximum likelihood methods and explicit in the Stein-type shrinkage approach. A simple version of a shrinkage estimator of the covariance matrix is represented by the Ledoit-Wolf shrinkage estimator.
Best linear unbiased predictions are similar to empirical Bayes estimates of random effects in linear mixed models, except that in the latter case, where weights depend on unknown values of components of variance, these unknown variances are replaced by sample-based estimates.
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.
Empirical Bayes; Hierarchical model; Posterior approximation; Markov chain Monte Carlo; Laplace's approximation; Integrated nested Laplace approximations; Variational inference; Approximate Bayesian computation; Estimators; Bayesian estimator; Credible interval; Maximum a posteriori estimation; Evidence approximation; Evidence lower bound ...
Consider a data set (,), …, (,), where the are Euclidean vectors and the are scalars.The multiple regression model is formulated as = +. where the are random errors. Zellner's g-prior for is a multivariate normal distribution with covariance matrix proportional to the inverse Fisher information matrix for , similar to a Jeffreys prior.