Monte Carlo swindles

While reading Boos and Hugues-Olivier’s 1998 American Statistician paper on the applications of Basu’s theorem I can across the notion of Monte Carlo swindles. Where a reduced variance can be achieved without the corresponding increase in Monte Carlo budget. For instance, approximating the variance of the median statistic Μ for a Normal location family can be sped up by considering that

\text{var}(M)=\text{var}(M-\bar X)+\text{var}(\bar X)

by Basu’s theorem. However, when reading the originating 1973 paper by Gross (although the notion is presumably due to Tukey), the argument boils down to Rao-Blackwellisation (without the Rao-Blackwell theorem being mentioned). The related 1985 American Statistician paper by Johnstone and Velleman exploits a latent variable representation. It also makes the connection with the control variate approach, noticing the appeal of using the score function as a (standard) control and (unusual) swindle, since its expectation is zero. I am surprised at uncovering this notion only now… Possibly because the method only applies in special settings.

A side remark from the same 1998 paper, namely that the enticing decomposition

\mathbb E[(X/Y)^k] = \mathbb E[X^k] \big/ \mathbb E[Y^k]

when X/Y and Y are independent, should be kept out of reach from my undergraduates at all costs, as they would quickly get rid of the assumption!!!

One Response to “Monte Carlo swindles”

  1. I had seen that term in Hamiltonian Monte Carlo Swindles by Dan Piponi, Matthew D. Hoffman, Pavel Sountsov a few years back:

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