The data augmentation algorithm : Theory and methodology

Assume that the function fX : R → [0,∞) is a probability density function (pdf). Suppose that g : R → R is a function of interest and that we want to know the value of EfXg = ∫ Rp g(x)fX(x) dx, but this integral cannot be computed analytically. There are many ways of approximating such intractable integrals and these include numerical integration, analytical approximations and Monte Carlo methods. In this chapter, we will describe a Markov chain Monte Carlo (MCMC) method called the data augmentation (DA) algorithm.

[1]  Djc MacKay,et al.  Slice sampling - Discussion , 2003 .

[2]  Xiao-Li Meng,et al.  Seeking efficient data augmentation schemes via conditional and marginal augmentation , 1999 .

[3]  J. Hobert,et al.  Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression , 2007 .

[4]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[5]  Jun S. Liu,et al.  Covariance Structure and Convergence Rate of the Gibbs Sampler with Various Scans , 1995 .

[6]  J. Hobert,et al.  Block Gibbs Sampling for Bayesian Random Effects Models With Improper Priors: Convergence and Regeneration , 2009 .

[7]  Galin L. Jones,et al.  On the applicability of regenerative simulation in Markov chain Monte Carlo , 2002 .

[8]  J. Rosenthal,et al.  Geometric Ergodicity and Hybrid Markov Chains , 1997 .

[9]  Ming-Hui Chen,et al.  Propriety of posterior distribution for dichotomous quantal response models , 2000 .

[10]  Adrian F. M. Smith,et al.  Simple conditions for the convergence of the Gibbs sampler and Metropolis-Hastings algorithms , 1994 .

[11]  J. Rosenthal.  Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo , 1995 .

[12]  J. Hobert,et al.  Geometric Ergodicity of van Dyk and Meng's Algorithm for the Multivariate Student's t Model , 2004 .

[13]  J. Hobert,et al.  A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms , 2008, 0804.0671.

[14]  E. Nummelin.  General irreducible Markov chains and non-negative operators: Embedded renewal processes , 1984 .

[15]  M. L. Eaton.  Group invariance applications in statistics , 1989 .

[16]  S. Chib,et al.  Bayesian analysis of binary and polychotomous response data , 1993 .

[17]  J. Rosenthal,et al.  Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains , 2006, math/0702412.

[18]  Gareth O. Roberts,et al.  Markov Chains and De‐initializing Processes , 2001 .

[19]  Jun S. Liu,et al.  Parameter Expansion for Data Augmentation , 1999 .

[20]  J. Rosenthal,et al.  General state space Markov chains and MCMC algorithms , 2004, math/0404033.

[21]  P. Diaconis,et al.  Gibbs sampling, exponential families and orthogonal polynomials , 2008, 0808.3852.

[22]  C. Robert.  Simulation of truncated normal variables , 2009, 0907.4010.

[23]  R. Tweedie,et al.  Geometric L 2 and L 1 convergence are equivalent for reversible Markov chains , 2001, Journal of Applied Probability.

[24]  Galin L. Jones,et al.  Fixed-Width Output Analysis for Markov Chain Monte Carlo , 2006, math/0601446.

[25]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[26]  Antonietta Mira,et al.  Ordering Monte Carlo Markov Chains , 1999 .

[27]  Bin Yu,et al.  Regeneration in Markov chain samplers , 1995 .

[28]  Jun S. Liu,et al.  Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes , 1994 .

[29]  S. Meyn,et al.  Computable Bounds for Geometric Convergence Rates of Markov Chains , 1994 .

[30]  Galin L. Jones,et al.  Honest Exploration of Intractable Probability Distributions via Markov Chain Monte Carlo , 2001 .

[31]  James P. Hobert,et al.  Norm Comparisons for Data Augmentation , 2007 .

[32]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[33]  P. Diaconis,et al.  Geometric Bounds for Eigenvalues of Markov Chains , 1991 .

[34]  M. Steel,et al.  Multivariate Student -t Regression Models : Pitfalls and Inference , 1999 .

[35]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[36]  Ruitao Liu,et al.  When is Eaton’s Markov chain irreducible? , 2007 .

[37]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.