Archive for umbrella sampling

reXing the bridge

Posted in Books, pictures, Statistics with tags , , , , , , , , , on April 27, 2021 by xi'an

As I was re-reading Xiao-Li  Meng’s and Wing Hung Wong’s 1996 bridge sampling paper in Statistica Sinica, I realised they were making the link with Geyer’s (1994) mythical tech report, in the sense that the iterative construction of α functions “converges to the `reverse logistic regression’  described in Geyer (1994) for the two-density cases” (p.839). Although they also saw the later as an “iterative” application of Torrie and Valleau’s (1977) “umbrella sampling” estimator. And cited Bennett (1976) in the Journal of Computational Physics [for which Elsevier still asks for $39.95!] as the originator of the formula [check (6)]. And of the optimal solution (check (8)). Bennett (1976) also mentions that the method fares poorly when the targets do not overlap:

“When the two ensembles neither overlap nor satisfy the above smoothness condition, an accurate estimate of the free energy cannot be made without gathering additional MC data from one or more intermediate ensembles”

in which case this sequence of intermediate targets could be constructed and, who knows?!, optimised. (This may be the chain solution discussed in the conclusion of the paper.) Another optimisation not considered in enough detail is the allocation of the computing time to the two densities, maybe using a bandit strategy to avoid estimating the variance of the importance weights first.

self-healing umbrella sampling

Posted in Kids, pictures, Statistics, University life with tags , , , , , , , on November 5, 2014 by xi'an

Ten days ago, Gersende Fort, Benjamin Jourdain, Tony Lelièvre, and Gabriel Stoltz arXived a study about an adaptive umbrella sampler that can be re-interpreted as a Wang-Landau algorithm, if not the most efficient version of the latter. This reminded me very much of the workshop we had all together in Edinburgh last June. And even more of the focus of the molecular dynamics talks in this same ICMS workshop about accelerating the MCMC exploration of multimodal targets. The self-healing aspect of the sampler is to adapt to the multimodal structure thanks to a partition that defines a biased sampling scheme spending time in each set of the partition in a frequency proportional to weights. While the optimal weights are the weights of the sets against the target distribution (are they truly optimal?! I would have thought lifting low density regions, i.e., marshes, could improve the mixing of the chain for a given proposal), those are unknown and they need to be estimated by an adaptive scheme that makes staying in a given set the less desirable the more one has visited it. By increasing the inverse weight of a given set by a factor each time it is visited. Which sounds indeed like Wang-Landau. The plus side of the self-healing umbrella sampler is that it only depends on a scale γ (and on the partition). Besides converging to the right weights of course. The downside is that it does not reach the most efficient convergence, since the adaptivity weight decreases in 1/n rather than 1/√n.

Note that the paper contains a massive experimental side where the authors checked the impact of various parameters by Monte Carlo studies of estimators involving more than a billion iterations. Apparently repeated a large number of times.

The next step in adaptivity should be about the adaptive determination of the partition, hoping for a robustness against the dimension of the space. Which may be unreachable if I judge by the apparent deceleration of the method when the number of terms in the partition increases.