The generalized Hierarchical Gaussian Filter

TL;DR

The paper introduces a generalized Hierarchical Gaussian Filter (gHGF) that unifies predictive coding and hierarchical Gaussian filtering by incorporating nonlinear value coupling, volatility coupling, and precision-based message passing. It reframes inference as modular networks of belief nodes that perform three basic computations per time step and exchange bottom-up prediction errors and top-down predictions, with additional signals for precision and volatility. The work provides formal update equations for value coupling, demonstrates modularity for constructing complex hierarchies, and presents open-source TAPAS implementations to enable flexible empirical analyses in computational psychiatry. By expanding HGF to include cross-level interactions and nonlinear couplings, it offers a versatile tool for modeling perception, learning, and uncertainty in health and disease. The practical impact lies in a scalable, interpretable Bayesian framework suitable for probing individual differences in belief updating and for integrating multiple uncertainty sources in neural and behavioral data.

Abstract

Hierarchical Bayesian models of perception and learning feature prominently in contemporary cognitive neuroscience where, for example, they inform computational concepts of mental disorders. This includes predictive coding and hierarchical Gaussian filtering (HGF), which differ in the nature of hierarchical representations. In this work, we present a new class of artificial neural networks that unifies computational principles of PC and HGFs. We extend the space of generative models underlying HGF to include a form of nonlinear hierarchical coupling between state values akin to predictive coding and artificial neural networks in general. We derive the update equations corresponding to this generalization of HGF and conceptualize them as connecting a network of (belief) nodes where parent nodes either predict the state of child nodes or their rate of change. This enables us to (1) create modular architectures with generic computational steps in each node of the network, and (2) disclose the hierarchical message passing implied by generalized HGF models and to compare this to comparable schemes under predictive coding. The practical advances of this work are twofold: on the one hand, our extension allows for a modular construction of ANNs of arbitrarily complex hierarchical structure under the general principles of HGF. On the other hand, by providing a highly flexible implementation of hierarchical Bayesian models available as open source software, it enables new types of empirical data analysis in computational psychiatry.

Figures (8)

• Figure 1: Effects of different coupling types within the generative model of the HGF. Upper row: Plots illustrate the effects of a state's total (constant) volatility (left), drift (middle) and mean (right) on the state's evolution over time steps. Each plot shows 30 simulated state trajectories per value of volatility/drift/mean for a continuous state performing a Gaussian random walk over 50 time steps. Lower row: Plots illustrate the effects of phasic changes in a state's volatility (left), drift (middle), and mean (right). Each plot shows the trajectory of the parent node and one simulated trajectory of the child node coupled to the parent via volatility/drift/mean coupling.
• Figure 2: An example of a generative model of sensory inputs with 11 hidden states and two observable outcomes. In this example, the volatility parents $x_{\check{a}}$ and $x_{\check{d}}$ share a value parent $x_f$, which represents a "global" or shared volatility state. Circles -- continuous states, squares -- binary states, observable outcomes -- shaded. Volatility coupling -- dashed lines, value coupling -- straight lines, links of outcomes to their hidden states -- double arrows.
• Figure 3: Comparing the flow of information in the generative model of the HGF with the implied belief network. A In the generative model, higher-level states influence the evolution of lower-level states (top-down information flow), either by affecting their mean (value coupling, left) or by changing their evolution rate (volatility coupling, right). B Representation of the message-passing within and between belief nodes as implied by the HGF's belief update equations. New observations cause a cascade of message-passing between nodes that includes bottom-up and top-down information flow. Higher-level beliefs send down their posteriors to inform lower-level predictions. Lower-level belief nodes send prediction errors and the precision of their own prediction bottom-up to drive higher-level belief updating. Within a node, we have placed separate units for the three computational steps that each node has to perform at a given time: the prediction step (green), the update step which results in a new posterior belief (blue), and the prediction error step (red). This message passing scheme generalizes across value and volatility coupling, although the specific messages passed along the connections as well as the computations within the nodes will depend on coupling type (see main text and Figures \ref{['fig:vapeSteps']} and \ref{['fig:vopeSteps']} for details).
• Figure 4: Message-passing for value coupling. Interactions of two nodes, node $a$ and its value parent node $b$, are shown during the three steps of a trial (Prediction step, left; Update step, middle; Prediction error step, right). The quantities that are being computed in each step are highlighted in white. Note that each step, we only show the computations for either the parent or the child node. Connections with arrowheads indicate positive (excitatory) influences, connections with circular heads indicate negative (inhibitory) influences. Arrows ending on units indicate additive influences, those ending on other arrows indicate multiplicative influences. Each HGF quantity that changes across trials is assigned its own unit. Parameters ($\alpha$, $\kappa$, $\lambda$, $\omega$ and $\rho$) determine connection strengths. For clarity, the volatility and drift nodes $\Omega$ and $P$ are only shown during the prediction step.
• Figure 5: Message-passing for volatility coupling. Interactions of two nodes, node $a$ and its volatility parent node $\check{a}$, are shown during the three steps of a trial (Prediction step, left; Update step, middle; Prediction error step, right). The quantities that are being computed in each step are highlighted in white. Logic of display as in figure \ref{['fig:vapeSteps']}.