Estimating Ratios of Normalizing Constants Using Linked Importance Sampling

Radford M. Neal, Dept. of Statistics and Dept. of Computer Science, University of Toronto

Ratios of normalizing constants for two distributions are needed in both Bayesian statistics, where they are used to compare models, and in statistical physics, where they correspond to differences in free energy. Two approaches have long been used to estimate ratios of normalizing constants. The `simple importance sampling' (SIS) or `free energy perturbation' method uses a sample drawn from just one of the two distributions. The `bridge sampling' or `acceptance ratio' estimate can be viewed as the ratio of two SIS estimates involving a bridge distribution. For both methods, difficult problems must be handled by introducing a sequence of intermediate distributions linking the two distributions of interest, with the final ratio of normalizing constants being estimated by the product of estimates of ratios for adjacent distributions in this sequence. Recently, work by Jarzynski, and independently by Neal, has shown how one can view such a product of estimates, each based on simple importance sampling using a single point, as an SIS estimate on an extended state space. This `Annealed Importance Sampling' (AIS) method produces an exactly unbiased estimate for the ratio of normalizing constants even when the Markov transitions used do not reach equilibrium. In this paper, I show how a corresponding `Linked Importance Sampling' (LIS) method can be constructed in which the estimates for individual ratios are similar to bridge sampling estimates. As a further elaboration, bridge sampling rather than simple importance sampling can be employed at the top level for both AIS and LIS, which sometimes produces further improvement. I show empirically that for some problems, LIS estimates are much more accurate than AIS estimates found using the same computation time, although for other problems the two methods have similar performance. Like AIS, LIS can also produce estimates for expectations, even when the distribution contains multiple isolated modes. AIS is related to the `tempered transition' method for handling isolated modes, and to a method for `dragging' fast variables. Linked sampling methods similar to LIS can be constructed that are analogous to tempered transitions and to this method for dragging fast variables, which may sometimes work better than those analogous to AIS.

Technical Report No. 0511, Dept. of Statistics, University of Toronto (November 2005), 37 pages: postscript, pdf.

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