Abstract
AbstractWe present an objective Bayes method for covariance selection in Gaussian multivariate regression models having a sparse regression and covariance structure, the latter being Markov with respect to a directed acyclic graph (DAG). Our procedure can be easily complemented with a variable selection step, so that variable and graphical model selection can be performed jointly. In this way, we offer a solution to a problem of growing importance especially in the area of genetical genomics (eQTL analysis). The input of our method is a single default prior, essentially involving no subjective elicitation, while its output is a closed form marginal likelihood for every covariate‐adjusted DAG model, which is constant over each class of Markov equivalent DAGs; our procedure thus naturally encompasses covariate‐adjusted decomposable graphical models. In realistic experimental studies, our method is highly competitive, especially when the number of responses is large relative to the sample size.
Highlights
Graphical models are a well-established tool in multivariate statistics
Our interest lies in a collection of q random variables whose joint distribution, having density with respect to a product measure, embodies a conditional independence structure which can be represented by a Directed Acyclic Graph (DAG)
We return to the scenario discussed in the Introduction, leading to covariate-adjusted graphical model selection, and to the response matrix Y introduced at the beginning of section 2
Summary
Graphical models are a well-established tool in multivariate statistics. They allow to simplify high-dimensional distributions, both in terms of computations and in terms of interpretation, on the basis of a graph representing independencies between variables. Geiger & Heckerman (2002) listed a set of assumptions on the collection of parameter priors (across DAGs) which permit their construction starting from a single parameter prior under a complete DAG (a DAG with all pairs of vertices directly connected) This represents a dramatic simplification because: i) the specification of only one distribution is required, while all the remaining priors are derived from this one; ii) the latter distribution is placed on an unconstrained parameter space describing the model with no independencies. The problem is usually formulated as one of joint estimation of multiple regression coefficients and a precision matrix, with the latter assumed to be Markov with respect to some graph Since these models are used in high-dimensional settings, both the regression and the covariance structure are assumed to be sparse.
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