On the invariance of conditioning procedures for the specification of prior distributions for nested DAG models

Given a prior distribution for a model , the prior information specified on a nested submodel by means of a conditioning procedure crucially depends on the parameterisation used to describe the model. Regression coefficients represent the most common parameterisation of Gaussian DAG models. Nevertheless, in the specification of prior distributions, invariance considerations lead to the use of different parameterisations of the model, depending on the required invariance class. In this paper we consider the problem of prior specification by conditioning on zero regression coefficients and show that also such a procedure satisfies the property of invariance with respect to a class of parameterisations and characterise such a class.     

Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

Abstract. We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor (BF), requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper‐parameter, which can be set to its minimal value. We show that our approach produces genuine BFs. The implied prior on the concentration matrix of any complete graph is a data‐dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models and show that in this case they coincide with those recently obtained using limiting versions of hyper‐inverse Wishart distributions as priors on the graph‐constrained covariance matrices.