Watch out!, Reiichiro Kawai has just published a survey on infinite variance Monte Carlo methods in Probability Surveys, which is most welcomed as this issue is customarily ignored by both the literature and the practitioners. Radford Neal‘s warning about the dangers of using the harmonic mean estimator of the evidence (as in Newton and Raftery 1996) is an illustration that remains pertinent to this day. In that sense, the survey relates to specific, earlier if recent attempts, such as Chatterjee and Diaconis (2015) or Vehtari et al (2015), with its Pareto correction.
In its recapitulation of the basics of Monte Carlo (closely corresponding to my own introduction of the topic in undergraduate classes), the paper indicates that the consistency of the variance estimator is enough to replace the true variance with its estimator and maintain the CLT. I have often if vaguely wondered at the impact (if any) a variance estimator with (itself) an infinite variance would have. A note to this effect appears at the end of Section 1.2. While being involved from the start, importance sampling has to wait till section 3.2 to be formally introduced. It is also interesting to note that the original result on the optimal importance variance being zero when the integrand is always positive (or negative) is extended here, by noting that a zero variance estimator can always be found by breaking the integrand f into its positive and negative parts, and using now two single samples for the respective integrals. I thus find Example 6 rather unhelpful, even though the entire literature contains such examples with no added value of formal optimal importance samplers. A comment at the end of Example 6 is opens the door to a short discussion of reparametrisation in simulation, a topic rarely discussed in the literature. The use of Rao-Blackwellization as a variance reduction technique that is open to switching from infinite to finite variance, is emphasised as well in Section 2.1.
In relation with a recent musing of mine during a seminar in Warwick, the novel part in the survey on the limited usefulness of control variate is of interest, even though one could predict that linear regression is not doing very well in infinite variance environments. Examples 8 and 9 are most helpful in this respect. It is similarly revealing if unsurprising that basic antithetic variables do not help. The warning about detecting or failing to detect infinite variance situations is well-received.
While theoretically correct, the final section about truncation limit is more exploratory, in that truncation can produce biased answers, whose magnitude is not assessed within the experiment.
![{}^{ [(\lambda_2+rx_1)(\lambda_1+rx_2)-r]\exp[-(\lambda_1x_1+\lambda_2x_2+rx_1x_2)]}](https://s0.wp.com/latex.php?latex=%7B%7D%5E%7B+%5B%28%5Clambda_2%2Brx_1%29%28%5Clambda_1%2Brx_2%29-r%5D%5Cexp%5B-%28%5Clambda_1x_1%2B%5Clambda_2x_2%2Brx_1x_2%29%5D%7D&bg=000000&fg=B0B0B0&s=0&c=20201002)
