arXiv subjects

James P. Hobert

Publications and source records attributed to James P. Hobert.

26 records · Page 2Linked to original sources

arXiv2012

On the Geometric Ergodicity of Two-Variable Gibbs Samplers

A Markov chain is geometrically ergodic if it converges to its in- variant distribution at a geometric rate in total variation norm. We study geo- metric ergodicity of deterministic and random scan versions of the two-variable Gibbs sampler. We give a sufficient condition which simultaneously guarantees both versions are geometrically ergodic. We also develop a method for simul- taneously establishing that both versions are subgeometrically ergodic. These general results allow us to characterize the convergence rate of two-variable Gibbs samplers in a particular family of discrete bivariate distributions.

math.ST↗

arXiv2012

A spectral analytic comparison of trace-class data augmentation algorithms and their sandwich variants

The data augmentation (DA) algorithm is a widely used Markov chain Monte Carlo algorithm that is easy to implement but often suffers from slow convergence. The sandwich algorithm is an alternative that can converge much faster while requiring roughly the same computational effort per iteration. Theoretically, the sandwich algorithm always converges at least as fast as the corresponding DA algorithm in the sense that $\Vert {K^*}\Vert \le \Vert {K}\Vert$, where $K$ and $K^*$ are the Markov operators associated with the DA and sandwich algorithms, respectively, and $\Vert\cdot\Vert$ denotes operator norm. In this paper, a substantial refinement of this operator norm inequality is developed. In particular, under regularity conditions implying that $K$ is a trace-class operator, it is shown that $K^*$ is also a positive, trace-class operator, and that the spectrum of $K^*$ dominates that of $K$ in the sense that the ordered elements of the former are all less than or equal to the corresponding elements of the latter. Furthermore, if the sandwich algorithm is constructed using a group action, as described by Liu and Wu [J. Amer. Statist. Assoc. 94 (1999) 1264--1274] and Hobert and Marchev [Ann. Statist. 36 (2008) 532--554], then there is strict inequality between at least one pair of eigenvalues. These results are applied to a new DA algorithm for Bayesian quantile regression introduced by Kozumi and Kobayashi [J. Stat. Comput. Simul. 81 (2011) 1565--1578].

math.ST↗

arXiv2012

Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modeling

The reversible Markov chains that drive the data augmentation (DA) and sandwich algorithms define self-adjoint operators whose spectra encode the convergence properties of the algorithms. When the target distribution has uncountable support, as is nearly always the case in practice, it is generally quite difficult to get a handle on these spectra. We show that, if the augmentation space is finite, then (under regularity conditions) the operators defined by the DA and sandwich chains are compact, and the spectra are finite subsets of $[0,1)$. Moreover, we prove that the spectrum of the sandwich operator dominates the spectrum of the DA operator in the sense that the ordered elements of the former are all less than or equal to the corresponding elements of the latter. As a concrete example, we study a widely used DA algorithm for the exploration of posterior densities associated with Bayesian mixture models [J. Roy. Statist. Soc. Ser. B 56 (1994) 363--375]. In particular, we compare this mixture DA algorithm with an alternative algorithm proposed by Frühwirth-Schnatter [J. Amer. Statist. Assoc. 96 (2001) 194--209] that is based on random label switching.

stat.ME↗

arXiv2008

Evaluation of Formal posterior distributions via Markov chain arguments

We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function $ϕ$ of a parameter when the loss is quadratic. If the posterior mean of $ϕ$ is admissible for all bounded $ϕ$, the posterior is strongly admissible. We give sufficient conditions for strong admissibility. These conditions involve the recurrence of a Markov chain associated with the estimation problem. We develop general sufficient conditions for recurrence of general state space Markov chains that are also of independent interest. Our main example concerns the $p$-dimensional multivariate normal distribution with mean vector $θ$ when the prior distribution has the form $g(\|θ\|^2) dθ$ on the parameter space $\mathbb{R}^p$. Conditions on $g$ for strong admissibility of the posterior are provided.

math.ST↗

arXiv2008

A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms

The data augmentation (DA) algorithm is a widely used Markov chain Monte Carlo (MCMC) algorithm that is based on a Markov transition density of the form $p(x|x')=\int_{\mathsf{Y}}f_{X|Y}(x|y)f_{Y|X}(y|x') dy$, where $f_{X|Y}$ and $f_{Y|X}$ are conditional densities. The PX-DA and marginal augmentation algorithms of Liu and Wu [J. Amer. Statist. Assoc. 94 (1999) 1264--1274] and Meng and van Dyk [Biometrika 86 (1999) 301--320] are alternatives to DA that often converge much faster and are only slightly more computationally demanding. The transition densities of these alternative algorithms can be written in the form $p_R(x|x')=\int_{\mathsf{Y}}\int _{\mathsf{Y}}f_{X|Y}(x|y')R(y,dy')f_{Y|X}(y|x') dy$, where $R$ is a Markov transition function on $\mathsf{Y}$. We prove that when $R$ satisfies certain conditions, the MCMC algorithm driven by $p_R$ is at least as good as that driven by $p$ in terms of performance in the central limit theorem and in the operator norm sense. These results are brought to bear on a theoretical comparison of the DA, PX-DA and marginal augmentation algorithms. Our focus is on situations where the group structure exploited by Liu and Wu is available. We show that the PX-DA algorithm based on Haar measure is at least as good as any PX-DA algorithm constructed using a proper prior on the group.

math.ST↗

arXiv2007

When is Eaton's Markov chain irreducible?

Consider a parametric statistical model $P(\mathrm{d}x|θ)$ and an improper prior distribution $ν(\mathrm{d}θ)$ that together yield a (proper) formal posterior distribution $Q(\mathrm{d}θ|x)$. The prior is called strongly admissible if the generalized Bayes estimator of every bounded function of $θ$ is admissible under squared error loss. Eaton [Ann. Statist. 20 (1992) 1147--1179] has shown that a sufficient condition for strong admissibility of $ν$ is the local recurrence of the Markov chain whose transition function is $R(θ,\mathrm{d}η)=\int Q(\mathrm{d}η|x)P(\mathrm {d}x|θ)$. Applications of this result and its extensions are often greatly simplified when the Markov chain associated with $R$ is irreducible. However, establishing irreducibility can be difficult. In this paper, we provide a characterization of irreducibility for general state space Markov chains and use this characterization to develop an easily checked, necessary and sufficient condition for irreducibility of Eaton's Markov chain. All that is required to check this condition is a simple examination of $P$ and $ν$. Application of the main result is illustrated using two examples.

math.ST↗

arXiv2004

A mixture representation of πwith applications in Markov chain Monte Carlo and perfect sampling

Let X={X_n:n=0,1,2,...} be an irreducible, positive recurrent Markov chain with invariant probability measure π. We show that if X satisfies a one-step minorization condition, then πcan be represented as an infinite mixture. The distributions in the mixture are associated with the hitting times on an accessible atom introduced via the splitting construction of Athreya and Ney [Trans. Amer. Math. Soc. 245 (1978) 493-501] and Nummelin [Z. Wahrsch. Verw. Gebiete 43 (1978) 309-318]. When the small set in the minorization condition is the entire state space, our mixture representation of πreduces to a simple formula, first derived by Breyer and Roberts [Methodol. Comput. Appl. Probab. 3 (2001) 161-177] from which samples can be easily drawn. Despite the fact that the derivation of this formula involves no coupling or backward simulation arguments, the formula can be used to reconstruct perfect sampling algorithms based on coupling from the past

math.PR↗

arXiv2004

Sufficient burn-in for Gibbs samplers for a hierarchical random effects model

We consider Gibbs and block Gibbs samplers for a Bayesian hierarchical version of the one-way random effects model. Drift and minorization conditions are established for the underlying Markov chains. The drift and minorization are used in conjunction with results from J. S. Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566] and G. O. Roberts and R. L. Tweedie [Stochastic Process. Appl. 80 (1999) 211-229] to construct analytical upper bounds on the distance to stationarity. These lead to upper bounds on the amount of burn-in that is required to get the chain within a prespecified (total variation) distance of the stationary distribution. The results are illustrated with a numerical example.

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