A Bayesian approach to computing posterior distribution and quantile functions

Abstract

Abstract We consider Bayesian techniques for estimating distribution functions and quantile functions from an intractable posterior density function f(·), which are efficient in the sense that relatively few evaluations of f(·) are required. This Bayesian quadrature approach is therefore appropriate when each function evaluation is very expensive. A Gaussian process model is assumed for the prior distribution of f(·), from which we can derive the posterior for f(·) and its integrals. Efficiency is achieved by incorporating prior information about the density function. In particular, we assume f(·) is reasonably well approximated by a normal density. The Bayesian model-based approach also allows us to select optimal evaluation points, and we use this to find an optimal set for estimating univariate distribution functions. If we have the additional information that f(·) integrates to 1, then we show how this information can also be modeled, and can produce more accurate results. The method is demonstrated by integrating a range of density functions with known distribution functions.