Methods for Approximating Integrals in Statistics with Special Emphasis on Bayesian Integration Problems

Abstract

This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed, giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling (and preferably adaptive importance sampling) based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature and, in particular, subregion adaptive integration are the algorithms of choice for lower-dimensional integrals. Due to the difficulties in assessing convergence to stationarity and the error in estimates, Markov chain methods are recommended only when there is no adequate alternative. In certain very high dimensional problems, however, Markov chain methods are the only hope. The importance of the parameterization of the integral is noted for the success of all the methods, and several useful reparameterizations are presented.