reXing the bridge

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.