Summary: I asked:
I'm trying to figure out what makes some Bayesian analyses "empirical" and
others not, but I haven't been successful. Can someone please enlighten me?
Thanks to the five who provided answers, which were:
1. See papers by Herbert Robbins (no relation to me) on Empirical Bayes;
e.g., The Empirical Bayes Approach to Statistical Decision Problems
published in the Annals of Mathematical Statistics in 1964.
Naomi B. Robbins
2. Because all basesian methods are "empirical", the term is nonsense.
Sometimes
it is used for methods which estimate the prior dist from the same data set
as
in the main analyses... A good introduction:
{Andrew Gelman and John B. Clarin and Hal S. Stern and Donald B. Rubin},
{Bayesian Data Anaysis}, 1997, {London: Chapman \& Hall}
Peter Malewski
3. In addition to the Gelman et al book, I'd also suggest looking at
Carlin and Louis (1996), Bayes and Empirical Bayes Methods for Data
Analysis,
NY, Chapman and Hall (now CRC Press).
The basic idea is this. When a prior is postulated, its distribution
often has one or more unknown parameters, called hyperparameters. One
can posit a hyperprior for these second-stage unknowns, and each of these
may have their own priors, etc. It doesn't take long to realize that
this approach gets ridiculously messy. An alternative strategy is to use
the data to provide estimates of the hyperparameters at some reasonable
stage, and to use the estimates in deriving a posterior distribution.
Clearly, there's more to the subject than this, but the above is what
separates `empirical' Bayes from `nonempirical' Bayes (for lack of a better
term).
The references cited above provide an excellent, in depth treatment of
the subject. You might also want to refer to a paper by George Casella
in American Statistician (1985, vol. 39, pp. 83--87) for a fine overview
of the subject. It may also be worth looking into the BUGS software
(http://www.mrc-bsu.cam.ac.uk/bugs), as it is essentially an implementation
of empirical Bayes methods using MCMC. The BUGS manuals and the Chapman &
Hall book by Gilks et al (1996), Markov Chain Monte Carlo in Practice,
show a different side of empirical Bayes inference than the other references
cited.
Dennis Murphy
4. As I understand it, empirical Bayes analyses estimate parameters in the
prior, and true Bayes analyses integrate hyperparameters over hyperpriors.
Brian D. Ripley
5. Empirical Bayes generally refers to using Bayes methods, but the data to
provide an empirical estimate of the prior distribution. The publisher
Chapman & Hall has a book titled something like, "Bayes and Empirical Bayes
....." (I've completely forgotten the author's name) which might be helpful.
Patrick Crockett
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