Selection of importance weights for monte carlo estimation of normalizing constants

This paper concerns the problem of estimating normalizing constants for multivariate densities. We first compare the well known technique of Importance Sampling (IS) with another technique that we call Importance Weighting (IW), which has been recently proposed by Gelfand and Dey (1994). Both techniques require the choice of a suitable density. Whereas it is quite well known that the asymptotic variance of an IS estimator is proportional to the chi-square divergence of the IS density w.r.t. the density of interest, we point out that for the asymptotic variance of the corresponding IW estimator the same results holds, except that the arguments of the divergence are interchanged. This suggests how to adapt to the problem of choosing an IW density procedures which have been already proposed for the choice of the IS density. In particular we show this feature for the algorithms proposed by Geweke (1989) and West (1993). The resulting procedures are illustrated with some examples.