On the Prior and Posterior Distributions Used in Graphical Modelling

Graphical model learning and inference are often performed using Bayesian techniques. In particular, learning is usually performed in two separate steps. First, the graph structure is learned from the data; then the parameters of the model are estimated conditional on that graph structure. While the probability distributions involved in this second step have been studied in depth, the ones used in the first step have not been explored in as much detail. In this paper, we will study the prior and posterior distributions defined over the space of the graph structures for the purpose of learning the structure of a graphical model. In particular, we will provide a characterisation of the behaviour of those distributions as a function of the possible edges of the graph. We will then use the properties resulting from this characterisation to define measures of structural variability for both Bayesian and Markov networks, and we will point out some of their possible applications.Comment: 28 pages, 6 figure

Learning Bayesian Networks with the bnlearn R Package

bnlearn is an R package which includes several algorithms for learning the structure of Bayesian networks with either discrete or continuous variables. Both constraint-based and score-based algorithms are implemented, and can use the functionality provided by the snow package to improve their performance via parallel computing. Several network scores and conditional independence algorithms are available for both the learning algorithms and independent use. Advanced plotting options are provided by the Rgraphviz package.Comment: 22 pages, 4 picture

Measures of Variability for Bayesian Network Graphical Structures

The structure of a Bayesian network includes a great deal of information about the probability distribution of the data, which is uniquely identified given some general distributional assumptions. Therefore it's important to study its variability, which can be used to compare the performance of different learning algorithms and to measure the strength of any arbitrary subset of arcs. In this paper we will introduce some descriptive statistics and the corresponding parametric and Monte Carlo tests on the undirected graph underlying the structure of a Bayesian network, modeled as a multivariate Bernoulli random variable. A simple numeric example and the comparison of the performance of some structure learning algorithm on small samples will then illustrate their use.Comment: 19 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:0909.168

An Empirical-Bayes Score for Discrete Bayesian Networks

Bayesian network structure learning is often performed in a Bayesian setting, by evaluating candidate structures using their posterior probabilities for a given data set. Score-based algorithms then use those posterior probabilities as an objective function and return the maximum a posteriori network as the learned model. For discrete Bayesian networks, the canonical choice for a posterior score is the Bayesian Dirichlet equivalent uniform (BDeu) marginal likelihood with a uniform (U) graph prior (Heckerman et al., 1995). Its favourable theoretical properties descend from assuming a uniform prior both on the space of the network structures and on the space of the parameters of the network. In this paper, we revisit the limitations of these assumptions; and we introduce an alternative set of assumptions and the resulting score: the Bayesian Dirichlet sparse (BDs) empirical Bayes marginal likelihood with a marginal uniform (MU) graph prior. We evaluate its performance in an extensive simulation study, showing that MU+BDs is more accurate than U+BDeu both in learning the structure of the network and in predicting new observations, while not being computationally more complex to estimate.Comment: 12 pages, PGM 201

Dirichlet Bayesian Network Scores and the Maximum Relative Entropy Principle

A classic approach for learning Bayesian networks from data is to identify a maximum a posteriori (MAP) network structure. In the case of discrete Bayesian networks, MAP networks are selected by maximising one of several possible Bayesian Dirichlet (BD) scores; the most famous is the Bayesian Dirichlet equivalent uniform (BDeu) score from Heckerman et al (1995). The key properties of BDeu arise from its uniform prior over the parameters of each local distribution in the network, which makes structure learning computationally efficient; it does not require the elicitation of prior knowledge from experts; and it satisfies score equivalence. In this paper we will review the derivation and the properties of BD scores, and of BDeu in particular, and we will link them to the corresponding entropy estimates to study them from an information theoretic perspective. To this end, we will work in the context of the foundational work of Giffin and Caticha (2007), who showed that Bayesian inference can be framed as a particular case of the maximum relative entropy principle. We will use this connection to show that BDeu should not be used for structure learning from sparse data, since it violates the maximum relative entropy principle; and that it is also problematic from a more classic Bayesian model selection perspective, because it produces Bayes factors that are sensitive to the value of its only hyperparameter. Using a large simulation study, we found in our previous work (Scutari, 2016) that the Bayesian Dirichlet sparse (BDs) score seems to provide better accuracy in structure learning; in this paper we further show that BDs does not suffer from the issues above, and we recommend to use it for sparse data instead of BDeu. Finally, will show that these issues are in fact different aspects of the same problem and a consequence of the distributional assumptions of the prior.Comment: 20 pages, 4 figures; extended version submitted to Behaviormetrik

Learning Bayesian Networks with the bnlearn R Package

bnlearn is an R package (R Development Core Team 2010) which includes several algorithms for learning the structure of Bayesian networks with either discrete or continuous variables. Both constraint-based and score-based algorithms are implemented, and can use the functionality provided by the snow package (Tierney et al. 2008) to improve their performance via parallel computing. Several network scores and conditional independence algorithms are available for both the learning algorithms and independent use. Advanced plotting options are provided by the Rgraphviz package (Gentry et al. 2010).