MCMSki [day 2]
I was still feeling poorly this morning with my brain in a kind of flu-induced haze so could not concentrate for a whole talk, which is a shame as I missed most of the contents of the astrostatistics session put together by David van Dyk… Especially the talk by Roberto Trotta I was definitely looking for. And the defence of nested sampling strategies for marginal likelihood approximations. Even though I spotted posterior distributions for WMAP and Plank data on the ΛCDM that reminded me of our own work in this area… Apologies thus to all speakers for dozing in and out, it was certainly not due to a lack of interest!
Sebastian Seehars mentioned emcee (for ensemble Monte Carlo), with a corresponding software nicknamed “the MCMC hammer”, and their own CosmoHammer software. I read the paper by Goodman and Ware (2010) this afternoon during the ski break (if not on a ski lift!). Actually, I do not understand why an MCMC should be affine invariant: a good adaptive MCMC sampler should anyway catch up the right scale of the target distribution. Other than that, the ensemble sampler reminds me very much of the pinball sampler we developed with Kerrie Mengersen (1995 Valencia meeting), where the target is the product of L targets,
and a Gibbs-like sampler can be constructed, moving one component (with index k, say) of the L-sample at a time. (Just as in the pinball sampler.) Rather than avoiding all other components (as in the pinball sampler), Goodman and Ware draw a single other component at random (with index j, say) and make a proposal away from it:
where ζ is a scale random variable with (log-) symmetry around 1. The authors claim improvement over a single track Metropolis algorithm, but it of course depends on the type of Metropolis algorithms that is chosen… Overall, I think the criticism of the pinball sampler also applies here: using a product of targets can only slow down the convergence. Further, the affine structure of the target support is not a given. Highly constrained settings should not cope well with linear transforms and non-linear reparameterisations would be more efficient….