Bayesian Models of Graphs, Arrays and Other Exchangeable Random Structures

TL;DR

This paper introduces Bayesian models for complex data structures like graphs and matrices, extending de Finetti's theorem, and discusses their applications in modern data analysis including network analysis and collaborative filtering.

Contribution

It generalizes de Finetti's theorem to non-sequence data structures and surveys Bayesian models for graphs and arrays with practical applications.

Findings

Generalization of de Finetti's theorem to graphs and matrices

Survey of Bayesian models for non-sequence data structures

Applications in network analysis and collaborative filtering

Abstract

The natural habitat of most Bayesian methods is data represented by exchangeable sequences of observations, for which de Finetti's theorem provides the theoretical foundation. Dirichlet process clustering, Gaussian process regression, and many other parametric and nonparametric Bayesian models fall within the remit of this framework; many problems arising in modern data analysis do not. This article provides an introduction to Bayesian models of graphs, matrices, and other data that can be modeled by random structures. We describe results in probability theory that generalize de Finetti's theorem to such data and discuss their relevance to nonparametric Bayesian modeling. With the basic ideas in place, we survey example models available in the literature; applications of such models include collaborative filtering, link prediction, and graph and network analysis. We also highlight connections to recent developments in graph theory and probability, and sketch the more general mathematical foundation of Bayesian methods for other types of data beyond sequences and arrays.