Poisson process model for Monte Carlo methods
Chris Maddison arXived today a paper on the use of Poisson processes in Monte Carlo simulation. based on the so-called Gumbel-max trick, which amounts to add to the log-probabilities log p(i) of the discrete target, iid Gumbel variables, and to take the argmax as the result of the simulation. A neat trick as it does not require the probability distribution to be normalised. And as indicated in the above quote to relate simulation and optimisation. The generalisation considered here replaces the iid Gumbel variates by a Gumbel process, which is constructed as an “exponential race”, i.e., a Poisson process with an exponential auxiliary variable. The underlying variates can be generated from a substitute density, à la accept-reject, which means this alternative bounds the true target. As illustrated in the plot above.
The paper discusses two implementations of the principle found in an earlier NIPS 2014 paper [paper that contains most of the novelty about this method], one that refines the partition and the associated choice of proposals, and another one that exploits a branch-and-bound tree structure to optimise the Gumbel process. With apparently higher performances. Overall, I wonder at the applicability of the approach because of the accept-reject structure: it seems unlikely to apply to high dimensional problems.
While this is quite exciting, I find it surprising that this paper completely omits references to Brian Ripley’s considerable input on simulation and point processes. As well as the relevant Geyer and Møller (1994). (I am obviously extremely pleased to see that our 2004 paper with George Casella and Marty Wells is quoted there. We had written this paper in Cornell, a few years earlier, right after the 1999 JSM in Baltimore, but it has hardly been mentioned since then!)