point process-based Monte Carlo
Clément Walter from Paris just pointed me to an arXived paper he had very recently gotten accepted for publication in Statistics and Computing. (Congrats!) Because his paper relates to nested sampling. And connects it with rare event simulation via interacting particle systems. And multilevel Monte Carlo. I had missed it when it came out on arXiv last December [as the title was unrelated with nested sampling if not Monte Carlo], but the paper brings fairly interesting new results about an ideal version of nested sampling that is
- unbiased when using an infinite number of terms;
- always better than the standard Monte Carlo estimator, variance-wise;
- connected with an implicit marked Poisson process; and
- enjoying a finite variance provided the quantity of interest has an 1+ε moment.
Of course, such results only hold for an ideal version and do not address the issue of the conditional simulations required by nested sampling. (Which has an impact on the computing time as the conditional simulation becomes more and more expensive as the likelihood value increases.) The explanation therein of the approximation of tail probabilities by a Poisson estimate makes the link with deterministic nested sampling much clearer to me. Point 2 above means that the nested sampling estimate always does better than the average of the likelihood values produced by an iid
or MCMC simulation from the prior distribution. The paper also borrows from the debiasing approach of Rhee and Glynn (already used by the Russian roulette) to turn truncated versions of the nested sampling estimator into an unbiased estimator, with a limited impact on the variance of the estimator. Truncation is associated with the generation of a geometric stopping time which parameter needs to be optimised. Without a more detailed reading, I am somewhat lost as to this optimisation remains feasible in complex settings… The paper contains an illustration for a Pareto distribution where optimisation and calibration can be conducted quite far. It also re-analyses the Mexican hat example of Skilling (2006), showing that our stopping rule may induce bias.