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. Predictive coding assumes that higher levels in a
given hierarchy influence the state (value) of lower levels. In HGF, however,
higher levels determine the rate of change at lower levels. Here, 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. We find that the algorithmic architecture instantiated by
the generalized HGF is largely compatible with that of predictive coding but
extends it with some unique predictions which arise from precision and
volatility related computations. Our developments enable highly flexible
implementations of hierarchical Bayesian models for empirical data analysis and
are available as open source software