Biometrika Advance Access originally published online on October 9, 2009
Biometrika 2009 96(4):821-834; doi:10.1093/biomet/asp049
Article |
Bayesian analysis of matrix normal graphical models
Department of Statistical Science, Duke University, Durham, North Carolina 27708, U.S.A. hao{at}stat.duke.edu mike{at}stat.duke.edu
Received for publication 1 February 2008. Revision received 1 February 2009.
We present Bayesian analyses of matrix-variate normal data with conditional independencies induced by graphical model structuring of the characterizing covariance matrix parameters. This framework of matrix normal graphical models includes prior specifications, posterior computation using Markov chain Monte Carlo methods, evaluation of graphical model uncertainty and model structure search. Extensions to matrix-variate time series embed matrix normal graphs in dynamic models. Examples highlight questions of graphical model uncertainty, search and comparison in matrix data contexts. These models may be applied in a number of areas of multivariate analysis, time series and also spatial modelling.
Key Words: Gaussian graphical model • Graphical model search • Hyper-inverse Wishart distribution • Marginal likelihood • Matrix normal model • Matrix-variate dynamic graphical model • Parameter expansion
References
-
Besag J. A candidate’s formula: a curious result in Bayesian prediction. Biometrika (1989) 76:183.
Carvalho C. M., Massam H., West M. Simulation of hyper-inverse Wishart distributions in graphical models. Biometrika (2007) 94:647–59.
Carvalho C. M., West M. Dynamic matrix-variate graphical models. Bayesian Anal. (2007a) 2:69–98.[CrossRef]
Carvalho C. M., West M. Dynamic matrix-variate graphical models—a synopsis. Bayesian Statistics, VIII—Bernardo J. M., Bayarri M. J., Berger J. O., Dawid A. P., Heckerman D., Smith A. F. M., West M., eds. (2007b) Oxford: Oxford University Press. 585–90.
Chib S. Marginal likelihood from the Gibbs output. J. Am. Statist. Assoc. (1995) 90:1313–21.[CrossRef][Web of Science]
Dawid A. P. Some matrix-variate distribution theory: notational considerations and a Bayesian application. Biometrika (1981) 68:265–74.
Dawid A. P., Lauritzen S. L. Hyper-Markov laws in the statistical analysis of decomposable graphical models. Ann. Statist. (1993) 21:1272–317.[CrossRef]
Dobra A., Jones B., Hans C., Nevins J., West M. Sparse graphical models for exploring gene expression data. J. Mult. Anal. (2004) 90:196–212.[CrossRef]
Dutilleul P. The MLE algorithm for the matrix normal distribution. J. Statist. Comp. Simul. (1999) 64:105–23.[CrossRef]
Finn J. D. A General Model for Multivariate Analysis (1974) New York: Holt, Rinehart and Winston.
Galecki A. General class of covariance structures for two or more repeated factors in longitudinal data analysis. Commun. Statist. (1994) A 23:3105–19.[CrossRef]
Gelman A. Parameterization and Bayesian modeling. J. Am. Statist. Assoc. (2004) 99:537–45.[CrossRef][Web of Science]
Gelman A. Prior distributions for variance parameters in hierarchical models. Bayesian Anal. (2006) 3:515–34.
Giudici P. Learning in graphical Gaussian models. Bayesian Statistics, 5—Bernado J. M., Berger J. O., Dawid A. P., Smith A. F. M., eds. (1996) Oxford: Oxford University Press. 621–28.
Giudici P., Green P. J. Decomposable graphical Gaussian model determination. Biometrika (1999) 86:785–801.
Gupta A. K., Nagar D. K. Matrix Variate Distributions (2000) Monographs and Surveys in Pure & Applied Mathematics 104. London: Chapman & Hall.
Hans C., Dobra A., West M. Shotgun stochastic search in regression with many predictors. J. Am. Statist. Assoc. (2007) 102:507–16.[CrossRef][Web of Science]
Hobert J. P., Marchev D. A theoretical comparison of the data augmentation, marginal augmentation and PXDA algorithms. Ann. Statist. (2008) 2:532–54.
Huizenga H. M., de Munck J. C., Waldorp L. J., Grasman R. Spatiotemporal EEG/MEG source analysis based on a parametric noisecovariance model. IEEE Trans. Biomed. Eng. (2002) 49:533–39.[CrossRef][Web of Science][Medline]
Jones B., Carvalho C. M., Dobra A., Hans C., Carter C., West M. Experiments in stochastic computation for high-dimensional graphical models. Statist. Sci. (2005) 20:388–400.[CrossRef]
Jordan M., Ghahramani Z., Jaakkola T., Saul L. An introduction to variational methods for graphical models. Mach. Learn. (1999) 37:183–233.[CrossRef]
Lauritzen S. L. Graphical Models (1996) Oxford: Clarendon Press.
Liu C., Rubin D. B., Wu Y. N. Parameter expansion to accelerate EM: the PX-EM algorithm. Biometrika (1998) 85:755–70.
Liu J. S., Wu Y. N. Parameter expansion for data augmentation. J. Am. Statist. Assoc. (1999) 94:1264–74.[CrossRef][Web of Science]
Mardia K. V., Goodall C. R. Spatial-temporal analysis of multivariate environmental monitoring data. In: Multivariate Environmental Statistics—Patil G. P., Rao C. R., eds. (1993) Amsterdam: Elsevier. 347–85.
McCulloch R. E., Polson N. G., Rossi P. E. Bayesian analysis of the multinomial probit model with fully identified parameters. J. Economet. (2000) 99:173–93.[CrossRef]
Mitchell M. W., Genton M. G., Gumpertz M. L. A likelihood ratio test for separability of covariances. J. Mult. Anal. (2006) 97:1025–43.[CrossRef]
Naik D. N., Rao S. S. Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix. J. Appl. Statist. (2001) 29:91–105.
Pole A., West M., Harrison P. J. Applied Bayesian Forecasting and Time Series Analysis (1994) New York: Chapman-Hall.
Quintana J. M., West M. Multivariate time series analysis: new techniques applied to international exchange rate data. Statistician (1987) 36:275–81.[CrossRef][Web of Science]
Roy V., Hobert J. P. Convergence rates and asymptotic standard errors for MCMC algorithms for Bayesian probit regression. J. R. Statist. Soc. (2007) B 69:607–23.[CrossRef]
Theobald D. L., Wuttke D. S. Empirical Bayes hierarchical models for regularizing maximum likelihood estimation in the matrix Gaussian procrustes problem. Proc. Nat. Acad. Sci. (2006) 103:18521–7.
West M., Harrison P. J. Bayesian Forecasting and Dynamic Models (1997) 2nd ed. New York: Springer.
Whittaker J. Graphical Models in Applied Multivariate Statistics (1990) Chichester, UK: John Wiley and Sons.
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