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We present a variational Bayesian identification procedure for polynomial NARMAX models based on message passing on a factor graph. Message passing allows us to obtain full posterior distributions for regression coefficients, precision parameters and noise instances by means of local computations distributed according to the factorization of the dynamic model. The posterior distributions are important to shaping the predictive distribution for outputs, and ultimately lead to superior model performance during 1-step ahead prediction and simulation.

Active inference (ActInf) is an emerging theory that explains perception and action in biological agents in terms of minimizing a free energy bound on Bayesian surprise. Goal-directed behavior is elicited by introducing prior beliefs on the underlying generative model. In contrast to prior beliefs, which constrain all realizations of a random variable, we propose an alternative approach through chance constraints, which allow for a (typically small) probability of constraint violation, and demonstrate how such constraints can be used as intrinsic drivers for goal-directed behavior in ActInf. We illustrate how chance-constrained ActInf weights all imposed (prior) constraints on the generative model, allowing, for example, for a trade-off between robust control and empirical chance constraint violation. Second, we interpret the proposed solution within a message passing framework. Interestingly, the message passing interpretation is not only relevant to the context of ActInf, but also provides a general-purpose approach that can account for chance constraints on graphical models. The chance constraint message updates can then be readily combined with other prederived message update rules without the need for custom derivations. The proposed chance-constrained message passing framework thus accelerates the search for workable models in general and can be used to complement message-passing formulations on generative neural models.

Model evidence is a fundamental performance measure in Bayesian machine learning as it represents how well a model fits an observed data set. Since model evidence is often an intractable quantity, the literature often resorts to computing instead the Bethe Free Energy (BFE), which for cyclefree models is a tractable upper bound on the (negative log-) model evidence. In this paper, we propose a different and faster evidence computation approach by tracking local normalization constants of sum-product messages, termed scale factors. We tabulate scale factor update rules for various elementary factor nodes and by experimental validation we verify the correctness of these update rules for models involving both discrete and continuous variables. We show how tracking scale factors leads to performance improvements compared to the traditional BFE computation approach.

The Gamma mixture model is a flexible probability distribution for representing beliefs about scale variables such as precisions. Inference in the Gamma mixture model for all latent variables is non-trivial as it leads to intractable equations. This paper presents two variants of variational message passing-based inference in a Gamma mixture model. We use moment matching and alternatively expectation-maximization to approximate the posterior distributions. The proposed method supports automated inference in factor graphs for large probabilistic models that contain multiple Gamma mixture models as plug-in factors. The Gamma mixture model has been implemented in a factor graph package and we present experimental results for both synthetic and real-world data sets.

In this paper we discuss variational message passing-based (VMP) inference in a switching Hierarchical Gaussian Filter (HGF). An HGF is a flexible hierarchical state space model that supports closed-form VMP-based approximate inference for tracking of both states and slowly time-varying parameters. Since natural signals often submit to regime-switching dynamics, there is a need for low-complexity closed-form inference in switching state space models. Here we extend the HGF model with parameter switching mechanics and derive closed-form VMP update rules for plug-in applications in factor graph-based models. These VMP rules support both tracking of latent variables and variational free energy as a model performance measure. We show that the switching HGF performs better than a non-switching HGF on modelling of a stock market data set.

This paper proposes a generative hierarchical probabilistic model for acoustic signals where both the frequency decomposition and log-power spectrum appear as latent variables. In order to facilitate efficient inference, we represent the model in a factor graph that includes a probabilistic Fourier transform and a Gaussian scale model as modules. We derive novel ways of performing variational message passing-based inference in the Gaussian scale model. As a result, in this model a probabilistic representation of the log-power spectrum of an acoustic signal can be effectively inferred online. The proposed model may find applications as a front end wherever probabilistic log-power spectral features of a signal are needed. We validate the model and message passing-based inference methods by tracking the log-power spectrum of a speech signal.

We address the problem of online state and parameter estimation in the Hierarchical Gaussian Filter (HGF), which is a multi-layer dynamic model with non-conjugate couplings between upper-layer hidden states and parameters of a lower layer. These non-conjugacies necessitate the approximation of marginalization and expectation integrals, while the online inference constraint renders batch learning and Monte Carlo sampling unsuitable. Here we formulate the problem as a message passing task on a factor graph and propose an online variational message passing-based state and parameter tracking algorithm, which uses Gaussian quadrature to deal with non-conjugacies. We present improved message update rules for all non-conjugate couplings, thus allowing a plug-in inference method for alternative models with equivalent non-conjugate layer couplings. The method is validated on a recorded time series of Bitcoin prices.

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.