  The Bayesian probit regression model (Albert and Chib (1993)) is popular and
widely used for binary regression. While the improper flat prior for the
regression coefficients is an appropriate choice in the absence of any prior
information, a proper normal prior is desirable when prior information is
available or in modern high dimensional settings where the number of
coefficients ($p$) is greater than the sample size ($n$). For both choices of
priors, the resulting posterior density is intractable and a Data Dugmentation
(DA) Markov chain is used to generate approximate samples from the posterior
distribution. Establishing geometric ergodicity for this DA Markov chain is
important as it provides theoretical guarantees for constructing standard
errors 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 is
geometrically 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 sufficient
conditions under which the DA Markov chain is trace-class, i.e., the
eigenvalues of the corresponding operator are summable. In particular, this
allows us to conclude that the Haar PX-DA sandwich algorithm (obtained by
inserting an inexpensive extra step in between the two steps of the DA
algorithm) is strictly better than the DA algorithm in an appropriate sense.
