## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 11, Number 1 (2017), 177-210.

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

Saptarshi Chakraborty and Kshitij Khare

#### Abstract

The Bayesian probit regression model (Albert and Chib [1]) 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 Augmentation (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.

#### Article information

**Source**

Electron. J. Statist., Volume 11, Number 1 (2017), 177-210.

**Dates**

Received: March 2016

First available in Project Euclid: 1 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1485939612

**Digital Object Identifier**

doi:10.1214/16-EJS1219

**Mathematical Reviews number (MathSciNet)**

MR3604022

**Zentralblatt MATH identifier**

1366.60093

**Subjects**

Primary: 60J05: Discrete-time Markov processes on general state spaces 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

Secondary: 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$

**Keywords**

Bayesian probit model binary regression geometric ergodicity proper normal prior trace class sandwich algorithms Data Augmentation Markov chain Monte Carlo

#### Citation

Chakraborty, Saptarshi; Khare, Kshitij. Convergence properties of Gibbs samplers for Bayesian probit regression with proper priors. Electron. J. Statist. 11 (2017), no. 1, 177--210. doi:10.1214/16-EJS1219. https://projecteuclid.org/euclid.ejs/1485939612