Predictive coding (PC) is an influential theory of information processing in the brain, providing a biologically plausible alternative to backpropagation. It is motivated in terms of Bayesian inference, as hidden states and parameters are optimised via gradient descent on variational free energy. However, implementations of PC rely on maximum \textit{a posteriori} (MAP) estimates of hidden states and maximum likelihood (ML) estimates of parameters, limiting their ability to quantify epistemic uncertainty. In this work, we investigate a Bayesian extension to PC that estimates a posterior distribution over network parameters. This approach, termed Bayesian Predictive coding (BPC), preserves the locality of PC and results in closed-form Hebbian weight updates. Compared to PC, our BPC algorithm converges in fewer epochs in the full-batch setting and remains competitive in the mini-batch setting. Additionally, we demonstrate that BPC offers uncertainty quantification comparable to existing methods in Bayesian deep learning, while also improving convergence properties. Together, these results suggest that BPC provides a biologically plausible method for Bayesian learning in the brain, as well as an attractive approach to uncertainty quantification in deep learning.

Predicting future trajectories of nearby objects, especially under occlusion, is a crucial task in autonomous driving and safe robot navigation. Prior works typically neglect to maintain uncertainty about occluded objects and only predict trajectories of observed objects using high-capacity models such as Transformers trained on large datasets. While these approaches are effective in standard scenarios, they can struggle to generalize to the long-tail, safety-critical scenarios. In this work, we explore a conceptual framework unifying trajectory prediction and occlusion reasoning under the same class of structured probabilistic generative model, namely, switching dynamical systems. We then present some initial experiments illustrating its capabilities using the Waymo open dataset.

An open question in the study of emergent behaviour in multi-agent Bayesian systems is the relationship, if any, between individual and collective inference. In this paper we explore the correspondence between generative models that exist at two distinct scales, using spin glass models as a sandbox system to investigate this question. We show that the collective dynamics of a specific type of active inference agent is equivalent to sampling from the stationary distribution of a spin glass system. A collective of specifically-designed active inference agents can thus be described as implementing a form of sampling-based inference (namely, from a Boltzmann machine) at the higher level. However, this equivalence is very fragile, breaking upon simple modifications to the generative models of the individual agents or the nature of their interactions. We discuss the implications of this correspondence and its fragility for the study of multiscale systems composed of Bayesian agents.

The exploration-exploitation trade-off is central to the description of adaptive behaviour in fields ranging from machine learning, to biology, to economics. While many approaches have been taken, one approach to solving this trade-off has been to equip or propose that agents possess an intrinsic 'exploratory drive' which is often implemented in terms of maximizing the agents information gain about the world -- an approach which has been widely studied in machine learning and cognitive science. In this paper we mathematically investigate the nature and meaning of such approaches and demonstrate that this combination of utility maximizing and information-seeking behaviour arises from the minimization of an entirely difference class of objectives we call divergence objectives. We propose a dichotomy in the objective functions underlying adaptive behaviour between \emph{evidence} objectives, which correspond to well-known reward or utility maximizing objectives in the literature, and \emph{divergence} objectives which instead seek to minimize the divergence between the agent's expected and desired futures, and argue that this new class of divergence objectives could form the mathematical foundation for a much richer understanding of the exploratory components of adaptive and intelligent action, beyond simply greedy utility maximization.

The Kalman filter is a fundamental filtering algorithm that fuses noisy sensory data, a previous state estimate, and a dynamics model to produce a principled estimate of the current state. It assumes, and is optimal for, linear models and white Gaussian noise. Due to its relative simplicity and general effectiveness, the Kalman filter is widely used in engineering applications. Since many sensory problems the brain faces are, at their core, filtering problems, it is possible that the brain possesses neural circuitry that implements equivalent computations to the Kalman filter. The standard approach to Kalman filtering requires complex matrix computations that are unlikely to be directly implementable in neural circuits. In this paper, we show that a gradient-descent approximation to the Kalman filter requires only local computations with variance weighted prediction errors. Moreover, we show that it is possible under the same scheme to adaptively learn the dynamics model with a learning rule that corresponds directly to Hebbian plasticity. We demonstrate the performance of our method on a simple Kalman filtering task, and propose a neural implementation of the required equations. The Bayesian Brain hypothesis has gained significant traction in cognitive neuroscience over the last two decades (Knill and Pouget, 2004; Doya et al., 2007; Pouget et al., 2013).