On Model Selection, Bayesian Networks, and the Fisher Information Integral

We study BIC-like model selection criteria and in particular, their refinements that include a constant term involving the Fisher information matrix. We perform numerical simulations that enable increasingly accurate approximation of this constant in the case of Bayesian networks. We observe that for complex Bayesian network models, the constant term is a negative number with a very large absolute value that dominates the other terms for small and moderate sample sizes. For networks with a fixed number of parameters, d, the leading term in the complexity penalty, which is proportional to d, is the same. However, as we show, the constant term can vary significantly depending on the network structure even if the number of parameters is fixed. Based on our experiments, we conjecture that the distribution of the nodes’ outdegree is a key factor. Furthermore, we demonstrate that the constant term can have a dramatic effect on model selection performance for small sample sizes.Peer reviewe

On Model Selection, Bayesian Networks, and the Fisher Information Integral

Abstract. We study BIC-like model selection criteria and in particular, their refinements that include a constant term involving the Fisher information matrix. We observe that for complex Bayesian network models, the constant term is a negative number with a very large absolute value that dominates the other terms for small and moderate sample sizes. We show that including the constant term degrades model selection accuracy dramatically compared to the standard BIC criterion where the term is omitted. On the other hand, we demonstrate that exact formulas such as Bayes factors or the normalized maximum likelihood (NML), or their approximations that are not based on Taylor expansions, perform well. A conclusion is that in lack of an exact formula, one should use either BIC, which is a very rough approximation, or a very close approximation but not an approximation that is truncated after the constant term.Peer reviewe

On Model Selection, Bayesian Networks, and the Fisher Information Integral

We study BIC-like model selection criteria and in particular, their refinements that include a constant term involving the Fisher information matrix. We perform numerical simulations that enable increasingly accurate approximation of this constant in the case of Bayesian networks. We observe that for complex Bayesian network models, the constant term is a negative number with a very large absolute value that dominates the other terms for small and moderate sample sizes. For networks with a fixed number of parameters, d, the leading term in the complexity penalty, which is proportional to d, is the same. However, as we show, the constant term can vary significantly depending on the network structure even if the number of parameters is fixed. Based on our experiments, we conjecture that the distribution of the nodes’ outdegree is a key factor. Furthermore, we demonstrate that the constant term can have a dramatic effect on model selection performance for small sample sizes.Peer reviewe

A Bayesian information criterion for singular models

We consider approximate Bayesian model choice for model selection problems that involve models whose Fisher-information matrices may fail to be invertible along other competing submodels. Such singular models do not obey the regularity conditions underlying the derivation of Schwarz's Bayesian information criterion (BIC) and the penalty structure in BIC generally does not reflect the frequentist large-sample behavior of their marginal likelihood. While large-sample theory for the marginal likelihood of singular models has been developed recently, the resulting approximations depend on the true parameter value and lead to a paradox of circular reasoning. Guided by examples such as determining the number of components of mixture models, the number of factors in latent factor models or the rank in reduced-rank regression, we propose a resolution to this paradox and give a practical extension of BIC for singular model selection problems

A Geometric Variational Approach to Bayesian Inference

We propose a novel Riemannian geometric framework for variational inference in Bayesian models based on the nonparametric Fisher-Rao metric on the manifold of probability density functions. Under the square-root density representation, the manifold can be identified with the positive orthant of the unit hypersphere in L2, and the Fisher-Rao metric reduces to the standard L2 metric. Exploiting such a Riemannian structure, we formulate the task of approximating the posterior distribution as a variational problem on the hypersphere based on the alpha-divergence. This provides a tighter lower bound on the marginal distribution when compared to, and a corresponding upper bound unavailable with, approaches based on the Kullback-Leibler divergence. We propose a novel gradient-based algorithm for the variational problem based on Frechet derivative operators motivated by the geometry of the Hilbert sphere, and examine its properties. Through simulations and real-data applications, we demonstrate the utility of the proposed geometric framework and algorithm on several Bayesian models

Hyper-g Priors for Generalized Linear Models

We develop an extension of the classical Zellner's g-prior to generalized linear models. The prior on the hyperparameter g is handled in a flexible way, so that any continuous proper hyperprior f(g) can be used, giving rise to a large class of hyper-g priors. Connections with the literature are described in detail. A fast and accurate integrated Laplace approximation of the marginal likelihood makes inference in large model spaces feasible. For posterior parameter estimation we propose an efficient and tuning-free Metropolis-Hastings sampler. The methodology is illustrated with variable selection and automatic covariate transformation in the Pima Indians diabetes data set.Comment: 30 pages, 12 figures, poster contribution at ISBA 201