new estimators of evidence

In an incredible accumulation of coincidences, I came across yet another paper about evidence and the harmonic mean challenge, by Yu-Bo Wang, Ming-Hui Chen [same as in Chen, Shao, Ibrahim], Lynn Kuo, and Paul O. Lewis this time, published in Bayesian Analysis. (Disclaimer: I was not involved in the reviews of any of these papers!)  Authors who arelocated in Storrs, Connecticut, in geographic and thematic connection with the original Gelfand and Dey (1994) paper! (Private joke about the Old Man of Storr in above picture!)

“The working parameter space is essentially the constrained support considered by Robert and Wraith (2009) and Marin and Robert (2010).”

The central idea is to use a more general function than our HPD restricted prior but still with a known integral. Not in the sense of control variates, though. The function of choice is a weighted sum of indicators of terms of a finite partition, which implies a compact parameter set Ω. Or a form of HPD region, although it is unclear when the volume can be derived. While the consistency of the estimator of the inverse normalising constant [based on an MCMC sample] is unsurprising, the more advanced part of the paper is about finding the optimal sequence of weights, as in control variates. But it is also unsurprising in that the weights are proportional to the inverses of the inverse posteriors over the sets in the partition. Since these are hard to derive in practice, the authors come up with a fairly interesting alternative, which is to take the value of the posterior at an arbitrary point of the relevant set.

The paper also contains an extension replacing the weights with functions that are integrable and with known integrals. Which is hard for most choices, even though it contains the regular harmonic mean estimator as a special case. And should also suffer from the curse of dimension when the constraint to keep the target almost constant is implemented (as in Figure 1).

The method, when properly calibrated, does much better than harmonic mean (not a surprise) and than Petris and Tardella (2007) alternative, but no other technique, on toy problems like Normal, Normal mixture, and probit regression with three covariates (no Pima Indians this time!). As an aside I find it hard to understand how the regular harmonic mean estimator takes longer than this more advanced version, which should require more calibration. But I find it hard to see a general application of the principle, because the partition needs to be chosen in terms of the target. Embedded balls cannot work for every possible problem, even with ex-post standardisation.

 

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