rediscovering the harmonic mean estimator

When looking at unanswered questions on X validated, I came across a question where the author wanted to approximate a normalising constant

N=\int g(x)\,\text{d}x\,,

while simulating from the associated density, g. While seemingly unaware of the (huge) literature in the area, he re-derived [a version of] the harmonic mean estimate by considering the [inverted importance sampling] identity

\int_\mathcal{X} \dfrac{\alpha(x)}{g(x)}p(x) \,\text{d}x=\int_\mathcal{X} \dfrac{\alpha(x)}{N} \,\text{d}x=\dfrac{1}{N}

when α is a probability density and by using for α the uniform over the whole range of the simulations from g. This choice of α obviously leads to an estimator with infinite variance when the support of g is unbounded, but the idea can be easily salvaged by using instead another uniform distribution, for instance on an highest density region, as we studied in our papers with Darren Wraith and Jean-Michel Marin. (Unfortunately, the originator of the question does not seem any longer interested in the problem.)

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