×

Trace-class Monte Carlo Markov chains for Bayesian multivariate linear regression with non-Gaussian errors. (English) Zbl 1426.62206

Summary: Let \(\pi\) denote the intractable posterior density that results when the likelihood from a multivariate linear regression model with errors from a scale mixture of normals is combined with the standard non-informative prior. There is a simple data augmentation algorithm (based on latent data from the mixing density) that can be used to explore \(\pi\). Let \(h\) and \(d\) denote the mixing density and the dimension of the regression model, respectively. J. P. Hobert et al. [Scand. J. Stat. 45, No. 3, 513–533 (2018; Zbl 1403.62132)] have recently shown that, if \(h\) converges to 0 at the origin at an appropriate rate, and \(\int_0^\infty u^{d/2} h(u) d u < \infty\), then the Markov chains underlying the data augmentation (DA) algorithm and an alternative Haar parameter expanded DA (PX-DA) algorithm are both geometrically ergodic. Their results are established using probabilistic techniques based on drift and minorization conditions. In this paper, spectral analytic techniques are used to establish that something much stronger than geometric ergodicity often holds. In particular, it is shown that, under simple conditions on \(h\), the Markov operators defined by the DA and Haar PX-DA Markov chains are trace-class, i.e., compact with summable eigenvalues. Many standard mixing densities satisfy the conditions developed in this paper. Indeed, the new results imply that the DA and Haar PX-DA Markov operators are trace-class whenever the mixing density is generalized inverse Gaussian, log-normal, Fréchet (with shape parameter larger than \(d/2\)), or inverted Gamma (with shape parameter larger than \(d/2\)).

MSC:

62J05 Linear regression; mixed models
60J22 Computational methods in Markov chains
62H12 Estimation in multivariate analysis
62F15 Bayesian inference

Citations:

Zbl 1403.62132
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adamczak, R.; Bednorz, W., Some remarks on MCMC estimation of spectra of integral operators, Bernoulli, 21, 2073-2092 (2015) · Zbl 1350.60027
[2] Ahues, M.; Largillier, A.; Limaye, B., Spectral Computations for Bounded Operators (2001), CRC Press: CRC Press London · Zbl 1053.47001
[3] Arnold, S. F., The Theory of Linear Models and Multivariate Analysis (1981), Wiley: Wiley New York
[4] Chan, K. S.; Geyer, C. J., Comment on “Markov chains for exploring posterior distributions” by L. Tierney, Ann. Statist., 22, 1747-1758 (1994)
[5] Conway, J. B., A Course in Functional Analysis (1990), Springer: Springer New York · Zbl 0706.46003
[6] Fernández, C.; Steel, M. F.J., Multivariate student-\(t\) regression models: Pitfalls and inference, Biometrika, 86, 153-167 (1999) · Zbl 0917.62020
[7] Flegal, J. M.; Haran, M.; Jones, G. L., Markov chain Monte Carlo: Can we trust the third significant figure?, Statist. Science, 23, 250-260 (2008) · Zbl 1327.62017
[8] Garren, S. T.; Smith, R. L., Estimating the second largest eigenvalue of a Markov transition matrix, Bernoulli, 6, 215-242 (2000) · Zbl 0976.62081
[9] Hobert, J. P.; Jung, Y. J.; Khare, K.; Qin, Q., Convergence analysis of MCMC algorithms for Bayesian multivariate linear regression with non-Gaussian errors, Scand. J. Stat. (2018) · Zbl 1403.62132
[10] Jones, G. L.; Hobert, J. P., Honest exploration of intractable probability distributions via markov chain Monte Carlo, Statist. Sci., 16, 312-334 (2001) · Zbl 1127.60309
[11] Jung, Y. J.; Hobert, J. P., Spectral properties of MCMC algorithms for Bayesian linear regression with generalized hyperbolic errors, Statist. Probab. Lett., 95, 92-100 (2014) · Zbl 1300.47111
[12] Khare, K.; Hobert, J. P., A spectral analytic comparison of trace-class data augmentation algorithms and their sandwich variants, Ann. Statist., 39, 2585-2606 (2011) · Zbl 1259.60081
[13] Koltchinskii, V.; Giné, E., Random matrix approximation of spectra of integral operators, Bernoulli, 6, 113-167 (2000) · Zbl 0949.60078
[14] Kontoyiannis, I.; Meyn, S. P., Geometric ergodicity and the spectral gap of non-reversible Markov chains, Probab. Theory Related Fields, 154, 327-339 (2012) · Zbl 1263.60064
[15] Liu, C., Bayesian robust multivariate linear regression with incomplete data, J. Amer. Statist. Assoc., 91, 1219-1227 (1996) · Zbl 0880.62028
[16] Liu, J. S.; Wong, W. H.; Kong, A., Covariance structure of the Gibbs sampler with applications to comparisons of estimators and augmentation schemes, Biometrika, 81, 27-40 (1994) · Zbl 0811.62080
[17] Q. Qin, J.P. Hobert, K. Khare, Estimating the spectral gap of a trace-class Markov operator, 2017, arXiv:1704.00850; Q. Qin, J.P. Hobert, K. Khare, Estimating the spectral gap of a trace-class Markov operator, 2017, arXiv:1704.00850
[18] Roberts, G. O.; Rosenthal, J. S., Geometric ergodicity and hybrid Markov chains, Electron. Commun. Probab., 2, 13-25 (1997) · Zbl 0890.60061
[19] Roberts, G. O.; Rosenthal, J. S., Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion), Canad. J. Statist., 26, 5-31 (1998) · Zbl 0920.62105
[20] Roberts, G. O.; Tweedie, R. L., Geometric \(L^2\) and \(L^1\) convergence are equivalent for reversible Markov chains, J. Appl. Probab., 38A, 37-41 (2001) · Zbl 1011.60050
[21] Roy, V.; Hobert, J. P., On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors, J. Multivariate Anal., 101, 1190-1202 (2010) · Zbl 1184.62040
[22] Weidmann, J., Linear Operators in Hilbert Spaces (2012), Springer: Springer New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.