Efficient estimation of the link function parameter in a robust Bayesian binary regression model

It is known that the robit regression model for binary data is a robust alternative to the more popular probit and logistic models. The robit model is obtained by replacing the normal distribution in the probit regression model with the Student’s t distribution. Unlike the probit and logistic models, the robit link has an extra degrees of freedom (df) parameter. It is shown that in practice it is important to estimate (rather than use a prespecified fixed value) the df parameter. A method for effectively selecting the df parameter of the robit model is described. The proposed method becomes computationally more effective if efficient MCMC algorithms are available for exploring the posterior distribution associated with a Bayesian robit model. Fast mixing parameter expanded DA (PX-DA) type algorithms based on an appropriate Haar measure are developed for significantly improving the convergence of DA algorithms for the robit model. The algorithms built for sampling from the Bayesian robit model shed new light on the construction of efficient PX-DA type algorithms in general. In spite of the fact that Haar PX-DA algorithms are known to be asymptotically “optimal”, through an empirical study it is shown that it may take millions of iterations before they provide improvement over the DA algorithms. Contrary to the popular belief, it is demonstrated that a partially reparameterized DA algorithm can outperform a fully reparameterized DA algorithm. The proposed methodology of selecting the df parameter is illustrated through two detailed examples.