Yesterday, I happened to be at the right time in the right place, as I was in Warwick for a RSS local section meeting on rare event simulation. (If missing the aurora borealis and the moon eclipse on previous nights!) And hence attended a seminar by Francesca Crucinio in six days!, as she talked about a turnkey approach to unbiased estimation of transforms of a moment, or wlog a mean μ, f(μ). A recent article with Nicolas Chopin (CREST) and Sumeet Singh, where they resort to Taylor expansions to achieve unbiasedness, using the Russian roulette trick to stop the summation from running to infinity. (As it happens, I heard Nicolas talk about this idea in the recent past namely at the ISBA-Fusion Sunday morn at Ca’Foscari.) Using a Taylor expansion is obviously natural and mathematically correct, albeit fraught with potential dangers [imho]:
- the Taylor expansion involves central moments up to a random order R, which are harder & harder to estimate with increasing orders (i.e., more & more uncertain, with the possibility of infinite variance estimators after a certain order)
- I did not spot a discussion on the moment estimators, that seems to rely on k iid replicas for the k-th moment
- a lot of calibration ensues, from the choice of the centre x⁰ to the (artificial) distribution of the stopping value R, to the parameterisation of the random variable attached to the moment μ
- the paper insists on recycling simulations to stabilise the moment estimators and ensure consistency, as a primary level of Rao-Blackwellisation, but this only applies to the smallest order moments and could be devised in many different ways, with varying computing costs
- consistency of the estimate is not necessarily needed, as for instance for pseudo-marginal applications
- as often with Russian roulette, positive quantities may receive negative estimations that are dominated by truncations to the positive real line (and alternating series offer the use of sandwiching estimators)
- for the above reason, it is not always reasonable to tunnel vision on unbiasedness and alternative estimates like bridge sampling solutions could be integrating towards improving the quality of the estimator (especially since the conditions for finite variance involve unknown quantities)
- while f-Taylored solutions like harmonic mean estimators for f(x)=1/x are not necessarily a panacea, they could be included in the comparison or as control variates
The first talk by Mathias Rousset was investigating adaptive multilevel sampling, a form of nested sampler, at the theoretical level, while the third talk by Tobias Grafke was a repetition of a talk he gave at the masterclass the interface between computational physics and computational statistics, last April.
