### Abstract

The use of MCMC algorithms in high dimensional Bayesian problems has become routine. This has spurred so-called convergence complexity analysis, the goal of which is to ascertain how the convergence rate of a Monte Carlo Markov chain scales with sample size, n, and/or number of covariates, p. This article provides a thorough convergence complexity analysis of Albert and Chib's [J. Amer. Statist. Assoc. 88 (1993) 669-679] data augmentation algorithm for the Bayesian probit regression model. The main tools used in this analysis are drift and minorization conditions. The usual pitfalls associated with this type of analysis are avoided by utilizing centered drift functions, which are minimized in high posterior probability regions, and by using a new technique to suppress high-dimensionality in the construction of minorization conditions. The main result is that the geometric convergence rate of the underlying Markov chain is bounded below 1 both as n → 8 (with p fixed), and as p → 8 (with n fixed). Furthermore, the first computable bounds on the total variation distance to stationarity are byproducts of the asymptotic analysis.

Original language | English (US) |
---|---|

Pages (from-to) | 2320-2347 |

Number of pages | 28 |

Journal | Annals of Statistics |

Volume | 47 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Drift condition
- Geometric ergodicity
- High dimensional inference
- Large p-small n
- Markov chain Monte Carlo
- Minorization condition

### Cite this

*Annals of Statistics*,

*47*(4), 2320-2347. https://doi.org/10.1214/18-AOS1749

**Convergence complexity analysis of albert and chib's algorithm for Bayesian probit regression.** / Qin, Qian; Hobert, James P.

Research output: Contribution to journal › Article

*Annals of Statistics*, vol. 47, no. 4, pp. 2320-2347. https://doi.org/10.1214/18-AOS1749

}

TY - JOUR

T1 - Convergence complexity analysis of albert and chib's algorithm for Bayesian probit regression

AU - Qin, Qian

AU - Hobert, James P.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The use of MCMC algorithms in high dimensional Bayesian problems has become routine. This has spurred so-called convergence complexity analysis, the goal of which is to ascertain how the convergence rate of a Monte Carlo Markov chain scales with sample size, n, and/or number of covariates, p. This article provides a thorough convergence complexity analysis of Albert and Chib's [J. Amer. Statist. Assoc. 88 (1993) 669-679] data augmentation algorithm for the Bayesian probit regression model. The main tools used in this analysis are drift and minorization conditions. The usual pitfalls associated with this type of analysis are avoided by utilizing centered drift functions, which are minimized in high posterior probability regions, and by using a new technique to suppress high-dimensionality in the construction of minorization conditions. The main result is that the geometric convergence rate of the underlying Markov chain is bounded below 1 both as n → 8 (with p fixed), and as p → 8 (with n fixed). Furthermore, the first computable bounds on the total variation distance to stationarity are byproducts of the asymptotic analysis.

AB - The use of MCMC algorithms in high dimensional Bayesian problems has become routine. This has spurred so-called convergence complexity analysis, the goal of which is to ascertain how the convergence rate of a Monte Carlo Markov chain scales with sample size, n, and/or number of covariates, p. This article provides a thorough convergence complexity analysis of Albert and Chib's [J. Amer. Statist. Assoc. 88 (1993) 669-679] data augmentation algorithm for the Bayesian probit regression model. The main tools used in this analysis are drift and minorization conditions. The usual pitfalls associated with this type of analysis are avoided by utilizing centered drift functions, which are minimized in high posterior probability regions, and by using a new technique to suppress high-dimensionality in the construction of minorization conditions. The main result is that the geometric convergence rate of the underlying Markov chain is bounded below 1 both as n → 8 (with p fixed), and as p → 8 (with n fixed). Furthermore, the first computable bounds on the total variation distance to stationarity are byproducts of the asymptotic analysis.

KW - Drift condition

KW - Geometric ergodicity

KW - High dimensional inference

KW - Large p-small n

KW - Markov chain Monte Carlo

KW - Minorization condition

UR - http://www.scopus.com/inward/record.url?scp=85072247959&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072247959&partnerID=8YFLogxK

U2 - 10.1214/18-AOS1749

DO - 10.1214/18-AOS1749

M3 - Article

AN - SCOPUS:85072247959

VL - 47

SP - 2320

EP - 2347

JO - Annals of Statistics

JF - Annals of Statistics

SN - 0090-5364

IS - 4

ER -