Compatible prior distributions for directed acyclic graph models

The application of certain Bayesian techniques, such as the Bayes factor and model averaging, requires the specification of prior distributions on the parameters of alternative models. We propose a new method for constructing compatible priors on the parameters of models nested in a given directed acyclic graph model, using a conditioning approach. We define a class of parameterizations that is consistent with the modular structure of the directed acyclic graph and derive a procedure, that is invariant within this class, which we name reference conditioning. Copyright 2004 Royal Statistical Society.

[1]  Joe Whittaker,et al.  The Isserlis matrix and its application to non-decomposable graphical Gaussian models , 1998 .

[2]  M. Frydenberg.  The chain graph Markov property , 1990 .

[3]  David Heckerman,et al.  Parameter Priors for Directed Acyclic Graphical Models and the Characteriration of Several Probability Distributions , 1999, UAI.

[4]  Piero Veronese,et al.  Order-invariant group reference priors for natural exponential families having a simple quadratic variance function , 2004 .

[5]  A. Dawid.  Some matrix-variate distribution theory: Notational considerations and a Bayesian application , 1981 .

[6]  J. Bernardo.  Reference Posterior Distributions for Bayesian Inference , 1979 .

[7]  E. Gutiérrez-Peña,et al.  Reference priors for exponential families , 2003 .

[8]  Steffen L. Lauritzen,et al.  Causal Inference from Graphical Models , 2001 .

[9]  Steffen L. Lauritzen,et al.  Compatible prior distributions , 2000 .

[10]  Guido Consonni,et al.  Conditionally Reducible Natural Exponential Families and Enriched Conjugate Priors , 2001 .

[11]  Nanny Wermuth,et al.  Multivariate Dependencies: Models, Analysis and Interpretation , 1996 .

[12]  Judea Pearl,et al.  Equivalence and Synthesis of Causal Models , 1990, UAI.

[13]  Guido Consonni,et al.  Enriched conjugate and reference priors for the Wishart family on symmetric cones , 2003 .

[14]  N. Wermuth.  Linear Recursive Equations, Covariance Selection, and Path Analysis , 1980 .

[15]  J. Pearl.  Causality: Models, Reasoning and Inference , 2000 .

[16]  D. Madigan,et al.  On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs , 1997 .

[17]  A. Roverato.  Hyper Inverse Wishart Distribution for Non-decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models , 2002 .

[18]  Malay Ghosh,et al.  ON THE INVARIANCE OF NONINFORMATIVE PRIORS , 1996 .

[19]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[20]  A. Roverato.  Cholesky decomposition of a hyper inverse Wishart matrix , 2000 .

[21]  David J. Spiegelhalter,et al.  Sequential updating of conditional probabilities on directed graphical structures , 1990, Networks.

[22]  P. Spirtes,et al.  Causation, prediction, and search , 1993 .

[23]  J. N. R. Jeffers,et al.  Graphical Models in Applied Multivariate Statistics. , 1990 .

[24]  Malay Ghosh,et al.  Some remarks on noninformative priors , 1995 .

[25]  David J. Spiegelhalter,et al.  Probabilistic Networks and Expert Systems , 1999, Information Science and Statistics.

[26]  David Maxwell Chickering,et al.  Learning Bayesian Networks: The Combination of Knowledge and Statistical Data , 1994, Machine Learning.

[27]  Adrian F. M. Smith,et al.  Conjugate Parameterizations for Natural Exponential Families , 1995 .

[28]  P. Maher,et al.  Handbook of Matrices , 1999, The Mathematical Gazette.

[29]  N. Wermuth.  Analogies between Multiplicative Models in Contingency Tables and Covariance Selection , 1976 .

[30]  Valentina Leucari,et al.  Compatible Priors for Causal Bayesian Networks , 2003 .