Exact inference using variable integrating constant importance distributions

In this paper a new class of importance distributions that are based on normal and t-distributions are introduced. They are labeled variable integrating constant distributions or VIC distributions. The main innovation in this class of distributions is that the standard error associated with each prameter is expanded in an exponential polynomial function of the scaled distance from its mode, with a different polynomial used in each direction and for each parameter. Resulting variances differ at each point in the parameter space, thereby breaking the flexibility limitations that are inherent in normal and t-distributions allowing them to be closely matched to the posterior distribution whose moments are the goal of the inference exercise. Two illustrations are presented in which the performance of normal, t, normal-VIC, and t-VIC importance distributions are compared. Results show the VIC distributions to produce globally stable weights and substantial improvements in the numerical efficiency of importance sampling and independence Metropolis estimates.