A Gibbs sampling approach to Bayesian analysis of generalized linear models for binary data (Q1389250)

Let \(y_{ij}\), \(i=1,\dots,n_i\), \(j=1,\dots,m\) be independent binary random observations which follow the Bernoulli distribution with \(\pi_i=P(y_{ij}=1)= F(x_i^t\beta)\), where \(\beta\) is the \((k+1)\)-dimensional vector of the unknown parameters and \(x_i^t=(1,x_{i_1},\dots,x_{i_k})\), where \(x_{ij}\) are values of the independent variables at the level \(i\). Following \textit{J.H. Albert} and \textit{S. Chib} [J. Am. Stat. Assoc. 88, No. 422, 669-679 (1993; Zbl 0774.62031)], the author suggested a modification of the Gibbs sampling approach to Bayesian inference on the unknown parameter \(\beta\) for an arbitrary link function \(F\). For this purpose he introduced the independent normal latent variables \(z_{ij}\), \(i=1,\dots,n_i\), \(j=1,\dots,m\), with \(N(x_i^t\beta,1)\) distribution that satisfies certain restrictions depending on the model and observations. Thus, under the multivariate normal prior on \(\beta\) all posterior conditional distributions are truncated normal with restrictions. An accurate method of approximation of restrictions by a set of piecewise linear functions is also proposed. It leads to a rather simple computational algorithm. Results of applications to logistic and complimentary log-log models are presented.