Editing Empirical Bayes method (section)

{{Short description|Bayesian statistical inference method in which the prior distribution is estimated from the data}}
{{Bayesian statistics}}

'''Empirical Bayes methods''' are procedures for  [[statistical inference]] in which the [[Prior probability|prior probability distribution]] is estimated from the data.  This approach stands in contrast to standard [[Bayesian probability|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 Bayes model|hierarchical model]] wherein the parameters at the highest level of the hierarchy are set to their most likely values, instead of being integrated out.<ref>{{cite book |first1=Bradley P. |last1=Carlin |first2=Thomas A. |last2=Louis |chapter=Empirical Bayes: Past, Present, and Future |pages=312–318 |title=Statistics in the 21st Century |editor-first=Adrian E. |editor-last=Raftery |editor2-first=Martin A. |editor2-last=Tanner |editor3-first=Martin T. |editor3-last=Wells |location= |publisher=Chapman & Hall |year=2002 |isbn=1-58488-272-7 }}</ref>  Empirical Bayes, also known as '''maximum [[marginal likelihood]]''',<ref name="Bishop05"/> represents a convenient approach for setting [[Hyperparameter (Bayesian statistics)|hyperparameters]], but has been mostly supplanted by fully Bayesian hierarchical analyses since the 2000s with the increasing availability of well-performing computation techniques.{{Citation needed|date=February 2025}} It is still commonly used, however, for variational methods in Deep Learning, such as [[Variational autoencoder|variational autoencoders]], where latent variable spaces are high-dimensional.