Objective Bayesian Search of Gaussian Directed Acyclic Graphical Models for Ordered Variables with Non‐Local Priors

Abstract

Directed acyclic graphical (DAG) models are increasingly employed in the study of physical and biological systems to model direct influences between variables. Identifying the graph from data is a challenging endeavor, which can be more reasonably tackled if the variables are assumed to satisfy a given ordering; in this case we simply have to estimate the presence or absence of each potential edge. Working under this assumption, we propose an objective Bayesian method for searching the space of Gaussian DAG models, which provides a rich output from minimal input. We base our analysis on non-local parameter priors, which are especially suited for learning sparse graphs, because they allow a faster learning rate, relative to ordinary local parameter priors, when the true unknown sampling distribution belongs to a simple model. We implement an efficient stochastic search algorithm, which deals effectively with data sets having sample size smaller than the number of variables, and apply our method to a variety of simulated and real data sets. Our approach compares favorably, in terms of the ROC curve for edge hit rate versus false alarm rate, to current state-of-the-art frequentist methods relying on the assumption of ordered variables; under this assumption it exhibits a competitive advantage over the PC-algorithm, which can be considered as a frequentist benchmark for unordered variables. Importantly, we find that our method is still at an advantage for learning the skeleton of the DAG, when the ordering of the variables is only moderately mis-specified. Prospectively, our method could be coupled with a strategy to learn the order of the variables, thus dropping the known ordering assumption.

Posterior graph selection and estimation consistency for high-dimensional Bayesian DAG models

“…The next result establishes strong selection consistency for the objective non-local prior based approach of [1]. The proof is provided in the Supplementary Document.…”

“…In [1], the authors present an alternative to the Wishart-based Bayesian framework for Gaussian DAG models by using non-local priors. Non-local priors were first introduced in [14] as densities that are identically zero whenever a model parameter is equal to its null value in the context of hypothesis testing (compared to local priors, which still preserve positive values at null parameter values).…”

“…In recent work, Altamore et al [1] develop a class of objective non-local priors for Gaussian DAG models. This class of priors is structurally different from the DAG Wishart priors of [6], and we also investigate posterior model selection consistency under these non-local priors.…”

Objective Bayesian Comparison of Constrained Analysis of Variance Models

“…In this work we have used a uniform prior on model space for the sake of simplicity and comparison with results obtained using alternative methods. Other choices for priors on model space can be used in conjunction with our method; see for instance Scott and Berger (2010) in the context of variable selection, or Carvalho and Scott (2009) and Altomare et al (2013) for graphical model determination. where α(θ, θ ′ |y) is the acceptance probability of the proposed value and q(θ, θ ′ |y)…”

Nonparametric Bayesian Inference in Biostatistics

“…There are recent developments in stochastic search algorithms other than MCMC, including shotgun stochastic search (SSS, Hans et al (2007)) and feature-inclusion stochastic search (FINCS, Scott & Carvalho (2008); Altomare et al (2013)). SSS evaluates many neighboring states (graphs) at each step in parallel and moves to a new state with a probability proportional to its marginal posterior.…”