A Bayesian Interpretation of the Internal Model Principle

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 ofinterpretation'', describing when a system can be interpreted as performing Bayesian filtering on an outside world in a consistent way.

A Bayesian Interpretation of the Internal Model Principle

The paper provides a comprehensive analysis of the Internal Model Principle (IMP) within the framework of Bayesian inference, illustrating a cross-disciplinary connection between control theory and cognitive science. It explores how controllers implement internal models to regulate systems and discusses whether these concepts can be formalized using Bayesian methodologies.

Overview of Internal Model Principle

The Internal Model Principle, traditionally rooted in control theory, posits that for effective regulation of a system, a controller must inherently possess a model of the system it intends to regulate. This principle has been pivotal in various domains, including artificial intelligence, biology, and cognitive science, supporting the notion that systems require internal models to predict and compensate for environmental disturbances.

The authors delineate the IMP's conventional formulations and assumptions, emphasizing its core idea that a controller capable of addressing dynamic inputs must contain a model of the disturbances it aims to mitigate. They examine the IMP through algebraic approaches, providing a precise definition that encompasses a wide array of systems without imposing specific geometric assumptions.

Bayesian Inference and Internal Models

In cognitive science, Bayesian inference frameworks have increasingly been applied to model cognitive processes, positing that agents (such as humans) act as Bayesian reasoners, constantly updating their beliefs about hidden states of the world based on new evidence. This aligns with the IMP by suggesting that similar computational structures could underpin both technical control systems and biological or cognitive agents.

The paper bridges these domains by proposing a formal connection between the internal models in IMP and Bayesian interpretations, using Markov categories to establish a formal framework. It argues that models within the IMP can be recast as a special case of possibilistic Bayesian filtering, wherein a deterministic system is interpreted as executing Bayesian updates. This interpretation uses categorical systems theory and advances synthetic approaches to probability.

Key Contributions

1. Formalization of Models in IMP: The paper formalizes a notion of "model" within the IMP using categorical systems, providing a more generalized definition that is not a priori connected to Bayesian reasoning but can be interpreted through this lens.
2. Bayesian Interpretation: It illustrates that the constructs of IMP admit a Bayesian filtering interpretation, thereby integrating deterministic control theory into a framework traditionally used for probabilistic inference.
3. Implication for Cognitive Science: By demonstrating this linkage, the paper implies a theoretical foundation for cognitive models that leverage Bayesian inference, suggesting that cognitive systems may indeed employ structures analogous to those in control systems for environmental interaction.

Practical and Theoretical Implications

The implications of this research span several domains:

• Control Theory: It enriches understanding within control systems, providing a framework that accommodates both deterministic and probabilistic interpretations, potentially improving robustness in control design.
• Cognitive Science: For cognitive science and AI, this framework supports constructing models that reflect how organisms anticipate, react, and interact with their surroundings using an internal model architecturally aligned with Bayesian inference.
• Future Developments: The research invites further exploration into how internal models can be applied in developing more sophisticated AI systems capable of autonomously adapting to changes in complex environments, particularly in areas like robotics and adaptive systems.

Ultimately, the paper offers a significant theoretical expansion by formalizing a bridge between deterministic control and probabilistic reasoning, potentially guiding future interdisciplinary advancements in adaptive systems and cognitive architectures.