A Measure-Theoretic Description of the Intrinsic View of Embodied Agents

We consider a general model of the sensori-motor loop of an agent interacting with the world. Here, we assume a particular causal structure, mechanistically described in terms of Markov kernels. In this generality, we define two σ-algebras of events in the world that describe two respective perspectives:  (1) the perspective of an external observer,  (2) the intrinsic perspective of the agent. Not all aspects of the world, seen from the external perspective, are accessible to the agent. This is expressed by the fact that the second σ-algebra is a subalgebra of the first one. We show that, under continuity and compactness assumptions, the global dynamics of the world can be simplified without changing the internal process. This simplification can serve as a minimal world model that the system must have in order to be consistent with the internal process.