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The Bayesian probit regression model (Albert and Chib (1993)) is popular andwidely used for binary regression. While the improper flat prior for theregression coefficients is an appropriate choice in the absence of any priorinformation, a proper normal prior is desirable when prior information isavailable or in modern high dimensional settings where the number ofcoefficients ($p$) is greater than the sample size ($n$). For both choices ofpriors, the resulting posterior density is intractable and a Data Dugmentation(DA) Markov chain is used to generate approximate samples from the posteriordistribution. Establishing geometric ergodicity for this DA Markov chain isimportant as it provides theoretical guarantees for constructing standarderrors for Markov chain based estimates of posterior quantities. In this paper,we first show that in case of proper normal priors, the DA Markov chain isgeometrically ergodic *for all* choices of the design matrix $X$, $n$ and $p$(unlike the improper prior case, where $n \geq p$ and another condition on $X$are required for posterior propriety itself). We also derive sufficientconditions under which the DA Markov chain is trace-class, i.e., theeigenvalues of the corresponding operator are summable. In particular, thisallows us to conclude that the Haar PX-DA sandwich algorithm (obtained byinserting an inexpensive extra step in between the two steps of the DAalgorithm) is strictly better than the DA algorithm in an appropriate sense.

Convergence properties of Gibbs samplers for Bayesian probit regression with proper priors