Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression

Consider a probit regression problem in which Y₁, [ellipsis (horizontal)], Yn are independent Bernoulli random variables such that [graphic removed] where xi is a p-dimensional vector of known covariates that are associated with Yi, β is a p-dimensional vector of unknown regression coefficients and...

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Bibliographic Details
Published in:	Journal of the Royal Statistical Society. Series B, Statistical methodology Vol. 69; no. 4; pp. 607 - 623
Main Authors:	Roy, Vivekananda, Hobert, James P
Format:	Journal Article
Language:	English
Published:	Oxford, UK Oxford, UK : Blackwell Publishing Ltd 01.09.2007 Blackwell Publishing Ltd Blackwell Publishers Blackwell Royal Statistical Society Oxford University Press
Series:	Journal of the Royal Statistical Society Series B
Subjects:	Acceleration of convergence Algorithms Asymptotic methods Asymptotic variance Bayesian analysis Bernoulli Hypothesis Central limit theorem Consistent estimators Convergence Data augmentation algorithm Density Efficiency Ergodic theory Errors Estimation methods Exact sciences and technology Gaussian distributions General topics Geometric ergodicity Lupus Markov analysis Markov chain Markov chains Markov processes Mathematical models Mathematics Minorization condition Monte Carlo simulation Normal distribution Numerical analysis Numerical analysis. Scientific computation Numerical methods in probability and statistics Perceptron convergence procedure Probability and statistics Probability theory and stochastic processes Probit models PX-DA algorithm Random variables Regeneration Regression analysis Reversible Markov chain Sciences and techniques of general use Simulation Standard error Statistical methods Statistical models Statistical variance Statistics Studies Variance Variance estimation PX-DA algorithm Gaussian distribution Geometric ergodicity Covariate Stochastic method Distribution function Simulation model Random variable Posterior distribution Monte Carlo method Existence theorem Convergence acceleration Prior distribution Asymptotic behavior Algorithm Computational complexity Statistical method Statistical regression Regression coefficient Numerical analysis Ergodic theorem Density estimation Second moment Statistical distribution Error estimation Error rate Non parametric estimation Asymptotic variance Asymptotic convergence Limit theorem Markov chain Data augmentation algorithm Convergence rate Approximation theory Bayes estimation Minorization condition Reversible Markov chain Covariance analysis Statistical moment Average Ergodicity Statistical estimation Variance analysis Mean estimation Independent variable Regeneration Central limit theorem Simulation
ISSN:	1369-7412, 1467-9868
Online Access:	Get full text
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Summary:	Consider a probit regression problem in which Y₁, [ellipsis (horizontal)], Yn are independent Bernoulli random variables such that [graphic removed] where xi is a p-dimensional vector of known covariates that are associated with Yi, β is a p-dimensional vector of unknown regression coefficients and Φ(·) denotes the standard normal distribution function. We study Markov chain Monte Carlo algorithms for exploring the intractable posterior density that results when the probit regression likelihood is combined with a flat prior on β. We prove that Albert and Chib's data augmentation algorithm and Liu and Wu's PX-DA algorithm both converge at a geometric rate, which ensures the existence of central limit theorems for ergodic averages under a second-moment condition. Although these two algorithms are essentially equivalent in terms of computational complexity, results of Hobert and Marchev imply that the PX-DA algorithm is theoretically more efficient in the sense that the asymptotic variance in the central limit theorem under the PX-DA algorithm is no larger than that under Albert and Chib's algorithm. We also construct minorization conditions that allow us to exploit regenerative simulation techniques for the consistent estimation of asymptotic variances. As an illustration, we apply our results to van Dyk and Meng's lupus data. This example demonstrates that huge gains in efficiency are possible by using the PX-DA algorithm instead of Albert and Chib's algorithm.
Bibliography:	http://dx.doi.org/10.1111/j.1467-9868.2007.00602.x ArticleID:RSSB602 ark:/67375/WNG-QMVRZGSL-W istex:FA04BA5D1AB979E2E9FC3B8356725608727B20ED SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23 ObjectType-Article-1 ObjectType-Feature-2
ISSN:	1369-7412 1467-9868
DOI:	10.1111/j.1467-9868.2007.00602.x