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.

Active inference is a corollary of the Free Energy Principle that prescribes how self-organizing biological agents interact with their environment. The study of active inference processes relies on the definition of a generative probabilistic model and a description of how a free energy functional is minimized by neuronal message passing under that model. This paper presents a tutorial introduction to specifying active inference processes by Forney-style factor graphs (FFG). The FFG framework provides both an insightful representation of the probabilistic model and a biologically plausible inference scheme that, in principle, can be automatically executed in a computer simulation. As an illustrative example, we present an FFG for a deep temporal active inference process. The graph clearly shows how policy selection by expected free energy minimization results from free energy minimization per se, in an appropriate generative policy model.

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.

Pure-tone audiometry—the process of estimating a person's hearing threshold from “audible” and “inaudible” responses to tones of varying frequency and intensity—is the basis for diagnosing and quantifying hearing loss. By taking a probabilistic modeling approach, both optimal tone selection (in terms of expected information gain) and hearing threshold estimation can be derived through Bayesian inference methods. The performance of probabilistic model-based audiometry methods is directly linked to the quality of the underlying model. In recent years, Gaussian process (GP) models have been shown to provide good results in this context. We present methods to improve the efficiency of GP-based audiometry procedures by improving the underlying model. Instead of a single GP, we propose to use a GP mixture model that can be conditioned on side-information about the subject. The underlying idea is that one can typically distinguish between different types of hearing thresholds, enabling a mixture model to better capture the statistical properties of hearing thresholds among a population. Instead of modeling all hearing thresholds by a single GP, a mixture model allows specific types of hearing thresholds to be modeled by independent GP models. Moreover, the mixing coefficients can be conditioned on side-information such as age and gender, capturing the correlations between age, gender, and hearing threshold. We show how a GP mixture model can be optimized for a specific target population by learning the parameters from a data set containing annotated audiograms. We also derive an optimal tone selection method based on greedy information gain maximization, as well as hearing threshold estimation through Bayesian inference. The proposed models are fitted to a data set containing roughly 176 thousand annotated audiograms collected in the Nordic countries. We compare the predictive accuracies of optimized mixture models of varying sizes with that of an optimized single-GP model. The usefulness of the optimized models is tested in audiometry simulations. Simulation results indicate that an optimized GP mixture model can significantly outperform an optimized single-GP model in terms of predictive accuracy, and leads to significant increases the efficiency of the resulting Bayesian audiometry procedure.

Active Inference (AIF) is a framework that can be used both to describe information processing in naturally intelligent systems, such as the human brain, and to design synthetic intelligent systems (agents). In this paper we show that Expected Free Energy (EFE) minimisation, a core feature of the framework, does not lead to purposeful explorative behaviour in linear Gaussian dynamical systems. We provide a simple proof that, due to the specific construction used for the EFE, the terms responsible for the exploratory (epistemic) drive become constant in the case of linear Gaussian systems. This renders AIF equivalent to KL control. From a theoretical point of view this is an interesting result since it is generally assumed that EFE minimisation will always introduce an exploratory drive in AIF agents. While the full EFE objective does not lead to exploration in linear Gaussian dynamical systems, the principles of its construction can still be used to design objectives that include an epistemic drive. We provide an in-depth analysis of the mechanics behind the epistemic drive of AIF agents and show how to design objectives for linear Gaussian dynamical systems that do include an epistemic drive. Concretely, we show that focusing solely on epistemics and dispensing with goal-directed terms leads to a form of maximum entropy exploration that is heavily dependent on the type of control signals driving the system. Additive controls do not permit such exploration. From a practical point of view this is an important result since linear Gaussian dynamical systems with additive controls are an extensively used model class, encompassing for instance Linear Quadratic Gaussian controllers. On the other hand, linear Gaussian dynamical systems driven by multiplicative controls such as switching transition matrices do permit an exploratory drive.