back to a correction of the harmonic mean estimator

In a 2009 JCGS paper, Peter Lenk proposed a bias correction of the harmonic mean estimator, which is somewhat surprising given that the estimator usually has no variance and hence that its consistency is purely formal, since no speed of convergence can be taken for granted. In particular, the conjugate Normal model serving as a motivation leads to an infinite variance. The author is however blaming the poor behaviour of the harmonic mean estimator on the overly concentrated support of the posterior distribution, despite having no reservation about the original identity (with standard notations)

$m(x)^{-1} = \int \dfrac{\pi(\theta|x)}{f(x|\theta)}\,\text d \theta$

but suggesting the corrected

$m(x)^{-1} = \int_A \dfrac{\pi(\theta|x)}{f(x|\theta)}\,\text d \theta\big/ \Pi(A)$

although this is only true when A is within the support of the posterior. (In which case it connects with our own 2009 correction.) Opting for a set A corresponding to a “simulation support” of the posterior with a very vague meaning, if somewhat connected with the nested sampling starting set.

This entry was posted on May 11, 2023 at 12:23 am and is filed under Books, Statistics with tags harmonic mean estimator, importance sampling, Journal of Computational and Graphical Statistics, MCMC algorithm, Monte Carlo Statistical Methods, nested sampling. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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