Empirical Bayes method

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. 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 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 ${\displaystyle p(\eta )\,}$.

Information about a particular quantity of interest ${\displaystyle \theta _{i}\;}$ therefore comes not only from the properties of those data ${\displaystyle y}$ that directly depend on it, but also from the properties of the population of parameters ${\displaystyle \theta \;}$ as a whole, inferred from the data as a whole, summarised by the hyperparameters ${\displaystyle \eta \;}$.

Using Bayes' theorem,

In general, this integral will not be tractable analytically or symbolically and must be evaluated by numerical methods. Stochastic (random) or deterministic approximations may be used. Example stochastic methods are Markov Chain Monte Carlo and Monte Carlo sampling. Deterministic approximations are discussed in quadrature.

Alternatively, the expression can be written as

and the final factor in the integral can in turn be expressed as