Population Monte Carlo for cosmologists

The population Monte Carlo (PMC) study that is the result of an ANR (ANR-05-BLAN-0283-04 “Ecosstat”) collaboration between cosmologists at the Institut d’Astrophysique de Paris (IAP) and statisticians from Telecom Paritech and Dauphine is finally out on arXiv! The main bulk of the work was shared by two postdocs from the project, Darren Wraith and Martin Kilbinger, who thus deserve most of the praise.

The first part of the paper consists in an intensive comparison of PMC (as implemented in this arXived paper to appear in Statistics and Computing) with an adaptive state-of-the-art MCMC algorithm. The artificial target used for this comparison is of the banana-shape version in dimension 10 as in the figure on the left, inspired from Haario, Saksman and Tamminen (Bernoulli, 2001). The figure illustrates the adaptation of the importance function made of 9 Student’s t distributions with evolving parameters. The estimates of representative functions of interest in this setting are significantly improved when compared with an MCMC evaluation based on the same number of simulations. We also found that this banana target can be quite challenging when the correlation coefficient involved gets too high, as any simulation method then has difficulties reaching the tails. The evaluation of PMC also introduces convergence monitoring via the effective sample size and the less standard perplexity

that corresponds to the exponential of the Shannon entropy. is less than 1, with a proximity to 1 indicating a good fit between the target and the importance function. Compared with MCMC methods, using the perplexity in importance sampling settings allows for a simple convergence diagnostic.

The second part of the paper focus on a non-trivial implementation of the PMC scheme when the target is the posterior distribution on the cosmological parameters corresponding to three important datasets, CMB (associated with the WMAP5 code), SNIa and cosmic shear. The likelihood corresponding to some of those datasets can be quite demanding in terms of computing time and producing a good approximation of the posterior with a small number of points is crucial. Compared with MCMC approaches, the importance sampling nature of the algorithm allows for a speed improvement of 100, thanks to the parallel processing of the simulation! Besides this massive time gain (from several days to a few hours), the agreement between our PMC and the standard MCMC solutions is excellent.