## empirically Bayesian [wISBApedia]

Posted in Statistics with tags , , , , , , , on August 9, 2021 by xi'an

Last week I was pointed out a puzzling entry in the “empirical Bayes” Wikipedia page. The introduction section indeed contains a description of an iterative simulation method that involves an hyperprior $$p(η)$$even though the empirical Bayes perspective does not involve an hyperprior.

While the entry is vague and lacks formulae

These suggest an iterative scheme, qualitatively similar in structure to a Gibbs sampler, to evolve successively improved approximations to $$p(θ$$$$∣$$$$y)$$ and $$p(η∣y)$$. First, calculate an initial approximation to $$p(θ∣y)$$ ignoring the $$η$$ dependence completely; then calculate an approximation to $$p(η$$$$|$$$$y)$$ based upon the initial approximate distribution of $$p(θ$$$$∣$$$$y)$$; then use this $$p(η$$$$∣$$$$y)$$ to update the approximation for $$p(θ$$$$∣$$$$y)$$; then update $$p(η$$$$∣$$$$y)$$; and so on.

it sounds essentially equivalent to a Gibbs sampler, possibly a multiple try Gibbs sampler (unless the author had another notion in mind, alas impossible to guess since no reference is included).

Beyond this specific case, where I think the entire paragraph should be erased from the “empirical Bayes” Wikipedia page, I discussed the general problem of some poor Bayesian entries in Wikipedia with Robin Ryder, who came with the neat idea of running (collective) Wikipedia editing labs at ISBA conferences. If we could further give an ISBA label to these entries, as a certificate of “Bayesian orthodoxy” (!), it would be terrific!