**C**olin Wei and Iain Murray arXived a new version of their paper on doubly-intractable distributions, which is to be presented at AISTATS. It builds upon the Russian roulette estimator of Lyne et al. (2015), which itself exploits the debiasing technique of McLeish et al. (2011) [found earlier in the physics literature as in Carter and Cashwell, 1975, according to the current paper]. Such an unbiased estimator of the inverse of the normalising constant can be used for pseudo-marginal MCMC, except that the estimator is sometimes negative and has to be so as proved by Pierre Jacob and co-authors. As I discussed in my post on the Russian roulette estimator, replacing the negative estimate with its absolute value does not seem right because a negative value indicates that the quantity is close to zero, hence replacing it with zero would sound more appropriate. Wei and Murray start from the property that, while the expectation of the importance weight is equal to the normalising constant, the expectation of the inverse of the importance weight converges to the inverse of the weight for an MCMC chain. This however sounds like an harmonic mean estimate because the property would also stand for any substitute to the importance density, as it only requires the density to integrate to one… As noted in the paper, the variance of the resulting Roulette estimator “will be high” or even infinite. Following Glynn et al. (2014), the authors build a coupled version of that solution, which key feature is to cut the higher order terms in the debiasing estimator. This does not guarantee finite variance or positivity of the estimate, though. In order to decrease the variance (assuming it is finite), backward coupling is introduced, with a Rao-Blackwellisation step using our 1996 Biometrika derivation. Which happens to be of lower cost than the standard Rao-Blackwellisation in that special case, O(N) versus O(N²), N being the stopping rule used in the debiasing estimator. Under the assumption that the *inverse* importance weight has finite expectation [wrt the importance density], the resulting backward-coupling Russian roulette estimator can be proven to be unbiased, as it enjoys a finite expectation. (As in the generalised harmonic mean case, the constraint imposes thinner tails on the importance function, which then hampers the convergence of the MCMC chain.) No mention is made of achieving finite variance for those estimators, which again is a serious concern due to the similarity with harmonic means…

## Archive for Peter Glynn

## Russian roulette still rolling

Posted in Statistics with tags AISTATS 2017, Biometrika, coupling, debiasing, doubly intractable problems, harmonic mean estimator, MCMC, MCMC algorithm, normalising constant, Peter Glynn, pseudo-marginal MCMC, Rao-Blackwellisation, Russian roulette on March 22, 2017 by xi'an## pitfalls of nested Monte Carlo

Posted in Books, pictures, Statistics, University life with tags AISTATS, Luc Devroye, Monte Carlo integration, nested sampling, Peter Glynn, Russian roulette on December 19, 2016 by xi'an**A** few days ago, Tom Rainforth, Robert Cornish, Hongseok Yang, and Frank Wood from Oxford have arXived a paper on the limitations of nested Monte Carlo. By nested Monte Carlo [not *nested sampling*], they mean Monte Carlo techniques used to evaluate the expectation of a non-linear transform of an expectation, which often call for plug-in resolution. The main result is that this expectation cannot be evaluated by an unbiased estimator. Which is only mildly surprising. I do wonder if there still exist series solutions à la Glynn and Rhee, as in the Russian roulette version. Which is mentioned in a footnote. Or specially tuned versions, as suggested by some techniques found in Devroye’s book where the expectation of the exponential of another expectation is considered… (The paper is quite short, which may be correlated with the format imposed by some machine-learning conference proceedings like AISTATS.)