Bayesian Analysis Exploiting Discretization

Numerically evaluation a posterior through discretization

Bayesian analysis is often hard to implement. While working on sequential experimental design for GLM, we suggested a simple implementation that numerically assesses the posterior in cases where the closed form of the Bayesian posterior is hard to formulate, exploiting a discretization of the prior. Our representation is reminiscent of importance sampling, in that we simulate the discrete points from the prior, and then weight by the ratio of the posterior to the prior, that is, by the likelihood.
We have used Bayesian analysis as a reliable method for analyzing GLM when there are only few observations. Using regression techniques for analyzing GLM with a small sample often leads to unreliable results, including large confidence ellipsoids and high bias. Furthermore, this method allows estimation of the model coefficients even when the number of observations is smaller than the number of coefficients.
For details, see section 3.1 of Hovav A. Dror and David M. Steinberg (2006). ?Sequential Experimental Design for Generalized Linear Models,? Technical Report RP-SOR-0607, Tel Aviv University, available at http://www.math.tau.ac.il/~dms/GLM_Design

Examples for utilization of the method are found in the examples provided for sequential experimental design, either on the website, and in a different file on MATLAB central file exchange. In addition, this file contains two examples which focuses only in this issue:

Binomial example, providing Bayesian Posterior Interval (?Credible Interval?): BayesianInterval.html, BayesianInterval.m, Screenshot: BaysianInterval.jpg
Bayesian Inference for a Poisson multivariate example: BayesianInferencePoisson.html, BayesianInferencePoisson.m, Screenshot: BayesianInferencePoisson.jpg

Most recent details are available at: http://www.math.tau.ac.il/~dms/GLM_Design