Belief Propagation, Bethe Approximation and Polynomials

Factor graphs are important models for succinctly representing probability distributions in machine learning, coding theory, and statistical physics. Several computational problems, such as computing marginals and partition functions, arise naturally when working with factor graphs. Belief propagation is a widely deployed iterative method for solving these problems. However, despite its significant empirical success, several questions regarding the correctness and efficiency of belief propagation remain open. The Bethe approximation is an optimization-based method for approximating the partition functions. While it is known that the stationary points of the Bethe approximation coincide with the fixed points of belief propagation, in general, the relation between the Bethe approximation and the partition function is not well understood. It has been observed that for a few classes of factor graphs, the Bethe approximation gives a lower bound to the partition function, which distinguishes them from the general case, where neither a lower bound nor an upper bound holds universally. This has been rigorously proved for permanents and for attractive graphical models. Here, we consider bipartite factor graphs over binary alphabet and show that if the local constraints satisfy a certain analytic property, the Bethe approximation is a lower bound to the partition function, generalizing an analogous inequality between the permanent and the Bethe permanent of a matrix with non-negative entries. We arrive at this result by viewing the factor graphs through the lens of polynomials, which allows us to reformulate the Bethe approximation as an optimization problem involving polynomials. The sufficient condition for our lower bound property to hold is inspired by the recent developments in the theory of real stable polynomials. We believe that this way of viewing factor graphs and its connection to real stability might lead to a better understanding of belief propagation and factor graphs in general.

I. Introduction

Several important classes of probability distributions studied in statistical physics, coding theory, and machine learning can be succinctly represented as factor graphs [32], [48]. Informally, they provide a succinct way to describe multivariate functions by specifying variables and relations between them in a form of a hypergraph [27]. In this context, of interest are the inference problem of estimating marginal probabilities of certain variables and the problem of estimating the partition function of such a factor graph. In computer vision one applies such inference primitives to learn about objects in a stage being captured by several cameras [20]. They are also essential components for decoding algorithms for low-density parity check codes [21], [42]. In statistical physics, factor graphs are used to model physical systems, for instance a set of particles – in such a setting the energy of a configuration is inversely proportional to the probability at which it occurs, thus intuitively, inference problems on such factor graphs correspond to learning “typical configurations” of a given system (see the book [32]).