Multivariate Bernoulli distribution

TL;DR

This paper introduces the multivariate Bernoulli distribution as a flexible model for graph structures with binary nodes, capable of capturing complex interactions and incorporating covariates, with applications in graphical inference.

Contribution

The paper develops the multivariate Bernoulli distribution framework, including its properties, comparison with existing models, and extensions with covariates and variable selection techniques.

Findings

Multivariate Bernoulli can model higher-order interactions.

Independence and uncorrelatedness are equivalent in this model.

Extensions include covariate inclusion and non-linear effects estimation.

Abstract

In this paper, we consider the multivariate Bernoulli distribution as a model to estimate the structure of graphs with binary nodes. This distribution is discussed in the framework of the exponential family, and its statistical properties regarding independence of the nodes are demonstrated. Importantly the model can estimate not only the main effects and pairwise interactions among the nodes but also is capable of modeling higher order interactions, allowing for the existence of complex clique effects. We compare the multivariate Bernoulli model with existing graphical inference models - the Ising model and the multivariate Gaussian model, where only the pairwise interactions are considered. On the other hand, the multivariate Bernoulli distribution has an interesting property in that independence and uncorrelatedness of the component random variables are equivalent. Both the marginal and conditional distributions of a subset of variables in the multivariate Bernoulli distribution still follow the multivariate Bernoulli distribution. Furthermore, the multivariate Bernoulli logistic model is developed under generalized linear model theory by utilizing the canonical link function in order to include covariate information on the nodes, edges and cliques. We also consider variable selection techniques such as LASSO in the logistic model to impose sparsity structure on the graph. Finally, we discuss extending the smoothing spline ANOVA approach to the multivariate Bernoulli logistic model to enable estimation of non-linear effects of the predictor variables.