A coalgebraic perspective on predictive processing

Abstract: Predictive processing and active inference posit that the brain is a system performing Bayesian inference on the environment. By virtue of this, a prominent interpretation of predictive processing states that the generative model (a POMDP) encoded by the brain synchronises with the generative process (another POMDP) representing the environment while trying to explain what hidden properties of the world generated its sensory input. In this view, the brain is thought to become a copy of the environment. This claim has however been disputed, stressing the fact that a structural copy, or isomorphism as it is at times invoked to be, is not an accurate description of this process since the environment is necessarily more complex than the brain, and what matters is not the capacity to exactly recapitulate the veridical causal structure of the world. In this work, we make parts of this counterargument formal by using ideas from the theory of coalgebras, an abstract mathematical framework for dynamical systems that brings together work from automata theory, concurrency theory, probabilistic processes and other fields. To do so, we cast generative model and process, in the form of POMDPs, as coalgebras, and use maps between them to describe a form of consistency that goes beyond mere structural similarity, giving the necessary mathematical background to describe how different processes can be seen as behaviourally, rather than structurally, equivalent, i.e. how they can be seen as emitting the same observations, and thus minimise prediction error, over time without strict assumptions about structural similarity. In particular, we will introduce three standard notions of equivalence from the literature on coalgebras, evaluating them in the context of predictive processing and identifying the one closest to claims made by proponents of this framework.