Accurate evaluation of Bayesian model evidence for a given data set is a fundamental problem in model development. Since evidence evaluations are usually intractable, in practice variational free energy (VFE) minimization provides an attractive alternative, as the VFE is an upper bound on negative model log-evidence (NLE). In order to improve tractability of the VFE, it is common to manipulate the constraints in the search space for the posterior distribution of the latent variables. Unfortunately, constraint manipulation may also lead to a less accurate estimate of the NLE. Thus, constraint manipulation implies an engineering trade-off between tractability and accuracy of model evidence estimation. In this paper, we develop a unifying account of constraint manipulation for variational inference in models that can be represented by a (Forney-style) factor graph, for which we identify the Bethe Free Energy as an approximation to the VFE. We derive well-known message passing algorithms from first principles, as the result of minimizing the constrained Bethe Free Energy (BFE). The proposed method supports evaluation of the BFE in factor graphs for model scoring and development of new message passing-based inference algorithms that potentially improve evidence estimation accuracy.

Bayesian inference realizes optimal information processing through a full commitment to reasoning by probability theory. The Bayesian framework is positioned at the core of modern AI technology for applications such as speech and image recognition and generation, medical analysis, robot navigation, and more. The framework describes how a rational agent should update its beliefs when new information is revealed by the agent's environment. Unfortunately, perfect Bayesian reasoning is generally intractable, since calculations of (often) very high-dimensional integrals are required for many models of interest. As a result, a number of numerical algorithms for approximating Bayesian inference have been developed and implemented in probabilistic programming packages. Successful methods include the Laplace approximation (Gelman et al., 2015), variants of Monte Carlo (MC) sampling (Salimans et al., n.d.), Variational Inference (VI) (Blei et al., 2017), Automatic-Differentiation Variational Inference (ADVI) (Kucukelbir et al., 2017), and Black-Box Variational Inference (BBVI) (Bamler & Mandt, 2017).

We introduce reactive message passing (RMP) as a framework for executing schedule-free, scalable, and, potentially, more robust message passing-based inference in a factor graph representation of a probabilistic model. RMP is based on the reactive programming style, which only describes how nodes in a factor graph react to changes in connected nodes. We recognize reactive programming as the suitable programming abstraction for message passing-based methods that improve robustness, scalability, and execution time of the inference procedure and are useful for all future implementations of message passing methods. We also present our own implementation ReactiveMP.jl, which is a Julia package for realizing RMP through minimization of a constrained Bethe free energy. By user-defined specification of local form and factorization constraints on the variational posterior distribution, ReactiveMP.jl executes hybrid message passing algorithms including belief propagation, variational message passing, expectation propagation, and expectation maximization update rules. Experimental results demonstrate the great performance of our RMP implementation compared to other Julia packages for Bayesian inference across a range of probabilistic models. In particular, we show that the RMP framework is capable of performing Bayesian inference for large-scale probabilistic state-space models with hundreds of thousands of random variables on a standard laptop computer.