On Bayesian Analysis of Generalized Linear Models Using the Jacobian Technique

Abstract

In this article, we obtain an estimator of the regression parameters for generalized linear models, using the Jacobian technique. We restrict ourselves to the natural exponential family for the response variable and choose the conjugate prior for the natural parameter. Using the Jacobian of transformation, we obtain the posterior distribution for the canonical link function and thereby obtain the posterior mode for the link. Under the full rank assumption for the covariate matrix, we then find an estimator for the regression parameters for the natural exponential family. Then the proposed estimator is specially derived for the Poisson model with log link function, and the binomial response model with the logit link function. We also discuss extensions to the binomial response model when covariates are all positive. Finally, an illustrative real-life example is given for the Poisson model with log link. In order to estimate the standard error of our estimators, we use the Bernstein-von Mises theorem. Finally, we compare the results using our Jacobian technique with a maximum likelihood estimates for the regression parameters.