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

Abstract.

Given a prior distribution for a model $\mathcal{M}$, the prior information specified on a nested submodel $\mathcal{M}^{0}$ 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.