Empirical Bayes

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.

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 ${\displaystyle y=\{y_{1},y_{2},\dots ,y_{n}\}}$ are assumed to be generated from an unobserved set of parameters ${\displaystyle \theta =\{\theta _{1},\theta _{2},\dots ,\theta _{n}\}}$ according to a probability distribution ${\displaystyle p(y\mid \theta )\,}$. In turn, the parameters ${\displaystyle \theta }$ can be considered samples drawn from a population characterised by hyperparameters ${\displaystyle \eta \,}$ according to a probability distribution ${\displaystyle p(\theta \mid \eta )\,}$. In the hierarchical Bayes model, though not in the empirical Bayes approximation, the hyperparameters ${\displaystyle \eta \,}$ are considered to be drawn from an unparameterized distribution ${\displaystyle p(\eta )\,}$.

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