Computational Methods Using a Bayesian Hierarchical Generalized Linear Model

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Abstract
Suppose we observe Y 1, …, YN , where Yi has the exponential density f(yi |θi ) = exp{ϕi ([yiθi – b(θi )]}c(yi, ϕi ). The parameters of interest are not the canonical parameters θi but the means μi = b′(θi ). In the usual generalized linear model (GLM) setup, suppose the means μ1, …, μN are believed to satisfy a specific p-dimensional GLM g(μi ) = xT iβ, where the link function g and the regression coefficients {xi } are known and the regression vector β is unknown. Two problems of interest are the assessment of the goodness of fit of the GLM and the estimation of the means μi . The approach to these problems is by the use of a Bayesian two-stage prior distribution, a generalization of a model used by Lindley and Smith (1972) in the normal mean-estimation problem. At the first stage of the model, we assign independent conjugate distributions to θ1, …, θN , where the prior means of the μi satisfy the GLM. There are p + 1 unknown hyperparameters in this specification, the elements of the regression parameter β and a precision parameter λ. At the second stage of the prior model, the hyperparameters β and λ are assigned noninformative distributions. The posterior distribution for the θi is expressible as π({θi } | data) = J π({θi } | data, β, λ)π{β, λ | data) dβ dλ. The posterior distribution of the θi given the hyperparameters β and λ is tractable, but the posterior distribution of β and λ is intractable. The focus of this article is on tractable accurate approximations to the posterior distribution of β and λ. These approximations give simple expressions for posterior moments of the hyperparameters λ and β and for posterior moments of the means μi . The accuracy of the approximate methods is investigated for binomial data in which the means are believed to satisfy a logistic model. A thorough evaluation of the methods is made for the simple exchangeable model. A more complicated logistic model is fit to data from Rosenberg (1962), and the usual frequentist inferences and inferences using the Bayesian hierarchical model are illustrated.

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