empirical Bayes (CHANCE)
Empirical Bayes methods can crudely be seen as the poor man’s Bayesian analysis. They start from a Bayesian modelling, for instance the parameterised prior
and then, instead of setting α to a specific value or of assigning an hyperprior to this hyperparameter α, as in a regular or a hierarchical Bayes approach, the empirical Bayes paradigm consists in estimating α from the data. Hence the “empirical” label. The reference model used for the estimation is the integrated likelihood (or conditional marginal)
which defines a distribution density indexed by α and thus allows for the use of any statistical estimation method (moments, maximum likelihood or even Bayesian!). A classical example is provided by the normal exchangeable sample: if
and μ can be estimated by the empirical average of the observations. The next step in an empirical Bayes analysis is to act as if α had not been estimated from the data and to conduct a regular Bayesian processing of the data with this estimated prior distribution. In the above normal example, this means estimating the θi‘s by
with the characteristic shrinkage (to the average) property of the resulting estimator (Efron and Morris, 1973).
“…empirical Bayes isn’t Bayes.” B. Efron (p.90)
While using Bayesian tools, this technique is outside of the Bayesian paradigm for several reasons: (a) the prior depends on the data, hence it lacks foundational justifications; (b) the prior varies with the data, hence it lacks theoretical validations like Walk’s complete class theorem; (c) the prior uses the data once, hence the posterior uses the data twice (see the vignette about this sin in the previous issue); (d) the prior relies of an estimator, whose variability is not accounted for in the subsequent analysis (Morris, 1983). The original motivation for the approach (Robbins, 1955) was more non-parametric, however it gained popularity in the 70’s and 80’s both in conjunction with the Stein effect and as a practical mean of bypassing complex Bayesian computations. As illustrated by Efron’s book, it recently met with renewed interest in connection with multiple testing.