Graphical models are used to describe the conditional independence relations in multivariate data. They have been used for a variety of problems, including log-linear models (Liu and Massam, 2006), network analysis (Holland and Leinhardt, 1981; Strauss and Ikeda, 1990; Wasserman and Pattison, 1996; Pattison and Wasserman, 1999; Robins et al., 1999), graphical Gaussian models (Roverato and Whittaker, 1998; Giudici and Green, 1999; Marrelec and Benali, 2006), and genetics (Dobra et al., 2004). A distribution that satisfies the conditional independence structure of a graph is Markov. A graphical model is a family of distributions that is restricted to be Markov with respect to a certain graph. In a Bayesian problem, one may specify a prior over the graphical model. Such a prior is called a hyper Markov law if the random marginals also satisfy the independence constraints. Previous work in this area includes (Dempster, 1972; Dawid and Lauritzen, 1993; Giudici and Green, 1999; Letac and Massam, 2007). We explore graphical models based on a non-parametric family of distributions, developed from Dirichlet processes.
References
Bush, C.A. and MacEachern, S.N. (1996). A semiparametric Bayesian model for randomized block designs. Biometrika, 83 275–285.
Carvalho, C., Massam, H. and West, M. (2007). Simulation of hyper-inverse Wishart distributions in graphical models. Biometrika, 94 647–659. http://ftp.stat.duke.edu/WorkingPapers/05-03.html.
Dawid, A.P. and Lauritzen, S.L. (1993). Hyper Markov laws in the statistical analysis of decomposable graphical models. The Annals of Statistics, 21 1272–1317.
Dempster, A. (1972). Covariance selection. Biometrics, 28 157–175.
Dobra, A., Hans, C., Jones, B., Nevins, J., Yao, G. and West, M. (2004). Sparse graphical models for exploring gene expression data. Journal of Multivariate Analysis, 90 196–212.
Escobar, M.D. and West, M. (1995). Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association, 90 577–588.
Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. The Annals of Statistics, 1 209–230.
Ghosh, J.K. and Ramamoorthi, R. (1995). Consistency of Bayesian inference for survival analysis with or without censoring. In Analysis of Censored Data (H. Koul and J. Deshpande, eds.). 95–104.
Giudici, P. and Green, P. (1999). Decomposable graphical gaussian model determination. Biometrika, 86 785–801.
Griffin, J.E. and Steel, M.F.J. (2006). Order-based dependent Dirichlet processes. Journal of the American Statistical Association, 101.
Holland, P.W. and Leinhardt, S. (1981). An exponential family of probability distributions. Journal of the American Statistical Association, 76 33–50.
Kim, Y. and Lee, J. (2001). On posterior consistency of survival models. The Annals of Statistics, 29 668–686.
Letac, G. and Massam, H. (2007). Wishart distributions for decomposable graphs. The Annals of Statistics, 35 1278–1323.
Liu, J. and Massam, H. (2006). The conjugate prior for discrete hierarchical log-linear models. http://www.citebase.org/abstract?id=oai:arXiv. org:math/0609100.
Marrelec, G. and Benali, H. (2006). Asymptotic Bayesian structure learning using graph supports for Gaussian graphical models. Journal of Multivariate Analysis, 97 1451–1466.
Pattison, P. and Wasserman, S. (1999). Logit models and logistic regression for social networks: II. multivariate relations. British Journal of Mathematical and Statistical Psychology, 52 169–193.
Pievatolo, A. and Rotondi, R. (2000). Analysing the interevent time distribution to identify seismicity phases: A Bayesian nonparametric approach to the multiple changepoint problem. Applied Statistics, 49 543–562.
Robins, G., Pattison, P. and Wasserman, S. (1999). Logit models and logistic regressions for social networks: III. valued relations. Psychometrika, 64 371–394.
Roverato, A. and Whittaker, J. (1998). The Isserlis matrix and its application to non-decomposable graphical Gaussian models. Biometrika, 85 711–725.
Schervish, M.J. (1995). Theory of Statistics. Springer-Verlag, New York.
Sethuraman, J. (1994). A constructive definition of Dirichlet measures. Statistica Sinica.
Speed, T. and Kiiveri, H. (1986). Gaussian Markov distributions over finite graphs. The Annals of Statistics, 14 138–150.
Strauss, D. and Ikeda, M. (1990). Pseudolikelihood estimation for social networks. Journal of the American Statistical Association, 85 204–212.
Susarla, V. and Ryzin, J.V. (1976). Nonparametric Bayesian estimation of survival curves from incomplete observations. Journal of the American Statistical Association, 71 897–902.
Wasserman, S. and Pattison, P. (1996). Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and p*. Psychometrika, 61 401–425.
