Laplace's Method Approximations for Probabilistic Inference in Belief Networks with Continuous Variables

TL;DR

This paper explores Laplace's method as an effective approximation technique for probabilistic inference in belief networks with continuous variables, highlighting its accuracy and potential applications.

Contribution

It introduces Laplace's method as a practical tool for approximate inference and model comparison in continuous-variable belief networks, discussing its advantages and limitations.

Findings

Approximate posterior moments with errors of O(n^-2)

Effective for computing Bayes factors in model selection

Comparable limitations to maximum likelihood methods

Abstract

Laplace's method, a family of asymptotic methods used to approximate integrals, is presented as a potential candidate for the tool box of techniques used for knowledge acquisition and probabilistic inference in belief networks with continuous variables. This technique approximates posterior moments and marginal posterior distributions with reasonable accuracy [errors are O(n^-2) for posterior means] in many interesting cases. The method also seems promising for computing approximations for Bayes factors for use in the context of model selection, model uncertainty and mixtures of pdfs. The limitations, regularity conditions and computational difficulties for the implementation of Laplace's method are comparable to those associated with the methods of maximum likelihood and posterior mode analysis.