A Bayesian Interpretation of the Internal Model Principle
Manuel Baltieri, Martin Biehl, Matteo Capucci, Nathaniel Virgo
TL;DR
This work formalizes a category-theoretic formulation of the Internal Model Principle (IMP) and develops a Markov-category-based Bayesian interpretation to connect control-theoretic models with Bayesian inference. It proves that, for autonomous IMP scenarios, a model in the IMP sense induces a Bayesian filtering interpretation, thereby unifying a classical control-theoretic notion with modern Bayesian reasoning frameworks. The results reveal that the IMP model is a special case of a more general Bayesian interpretation, while also showing the limitations of the IMP notion in capturing full probabilistic Bayesian updates. The approach provides a principled bridge between cybernetics, control theory, and cognitive-science perspectives on environment and disturbance rejection, with potential implications for designing agents that reason about and adapt to environments. The work suggests avenues for extending Bayesian interpretations beyond possibilistic or approximate updates to richer probabilistic models of interaction and learning.
Abstract
The internal model principle, originally proposed in the theory of control of linear systems, nowadays represents a more general class of results in control theory and cybernetics. The central claim of these results is that, under suitable assumptions, if a system (a controller) can regulate against a class of external inputs (from the environment), it is because the system contains a model of the system causing these inputs, which can be used to generate signals counteracting them. Similar claims on the role of internal models appear also in cognitive science, especially in modern Bayesian treatments of cognitive agents, often suggesting that a system (a human subject, or some other agent) models its environment to adapt against disturbances and perform goal-directed behaviour. It is however unclear whether the Bayesian internal models discussed in cognitive science bear any formal relation to the internal models invoked in standard treatments of control theory. Here, we first review the internal model principle and present a precise formulation of it using concepts inspired by categorical systems theory. This leads to a formal definition of ``model'' generalising its use in the internal model principle. Although this notion of model is not a priori related to the notion of Bayesian reasoning, we show that it can be seen as a special case of possibilistic Bayesian filtering. This result is based on a recent line of work formalising, using Markov categories, a notion of ``interpretation'', describing when a system can be interpreted as performing Bayesian filtering on an outside world in a consistent way.