## Diffusive nested sampling

Posted in Statistics with tags , , , , on December 20, 2009 by xi'an

A paper on a new version of nested sampling, written by three physicists, has been posted on arXiv last week. Its purpose is the same as MultiNest, namely to overcome exploration difficulties due to multimodality. The core idea is to allow the likelihood climb that characterises nested sampling to backtrack by allowing for decreases in the likelihood at each iteration. This method works as a natural tempering, defining levels of likelihood exceedances, each within a theoretical fraction $e^{-1}$ of the previous level coverage, so that the nested sampling sequence may leave isolated but attractive modes. It can also be seen as a special case of stratified sampling, where the strata are defined in terms of the likelihood levels. However, since the move from one level j to another k seems to be defined in terms of the respective numbers of visits to both levels, $n(j)$ and $n(k)$, from the start of the algorithm, I wonder how this non-Markovian feature impacts the validity of the resulting Metropolis-Hastings move. (Wang and Landau, 2001, is mentioned, but this does not seem enough of an argument.) Furthermore, since the strata are estimated during the simulation, the additional estimation of the probability of each stratum may jeopardise the variance reduction offered by a traditional statification (see MCSM, Exercise 4.22). Overall, this is an interesting paper, in that it straightens out the connection of nested sampling with importance sampling (see my post-MaxEnt09 post), but the fact that it completely relies on an MCMC scheme makes me wonder (as for classical nested sampling) why the comparison with other importance or MCMC approximations of the evidence $\mathfrak{Z}$ is not conducted.