Bayesian analysis is
often hard to implement. While working on sequential
experimental design, 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. See section 3.1 of the technical
report for details.

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.

Examples for utilization of the method are found in the examples provided for sequential experimental design. In addition, here are 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

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