Towards a Geometry and Analysis for Bayesian Mechanics

Abstract

In this paper, a simple case of Bayesian mechanics under the free energy principle is formulated in axiomatic terms. We argue that any dynamical system with constraints on its dynamics necessarily looks as though it is performing inference against these constraints, and that in a non-isolated system, such constraints imply external environmental variables embedding the system. Using aspects of classical dynamical systems theory in statistical mechanics, we show that this inference is equivalent to a gradient ascent on the Shannon entropy functional, recovering an approximate Bayesian inference under a locally ergodic probability measure on the state space. We also use some geometric notions from dynamical systems theory-namely, that the constraints constitute a gauge degree of freedom-to elaborate on how the desire to stay self-organised can be read as a gauge force acting on the system. In doing so, a number of results of independent interest are given. Overall, we provide a related, but alternative, formalism to those driven purely by descriptions of random dynamical systems, and take a further step towards a comprehensive statement of the physics of self-organisation in formal mathematical language.

The inner screen model of consciousness: applying the free energy principle directly to the study of conscious experience

“…The FEP is a scientific or mathematical principle that, much like the principle of least action or the principle of maximum entropy, can be used to derive the mechanics of coupled random dynamical systems (Friston, 2019a;Ramstead et al, 2023). Just as one can think of the principle of least action as the variational principle that underwrites classical mechanics, one can think of the FEP as the variational principle that describes the way that beliefs evolve over time; namely, Bayesian mechanics (Ramstead et al, 2023;Sakthivadivel, 2022b).…”

“…Intuitively, for some given thing to be or act as a map of some other thing (considered as a territory), it must be, at least in some minimal sense, distinguishable or separable from that territory. This intuition is formalised in Bayesian mechanics with the construct of Markov blankets (Ramstead et al, 2023;Sakthivadivel, 2022b). The core idea is that systems that conform to the FEP are endowed with a boundary, called a Markov blanket, which separates a "thing" or particle from its environment-but also couples the one to the other .…”

“…That said, this form of mapping relates to "tracking" in the conventional sense, since internal paths of least action encode expectations about external paths (where action is the path integral of variational free energy). Crucially, the factors of the conditional density (e.g., types of objects, outcomes, and events) that "carve up" the world may vary arbitrarily, subject only to the constraint that model evidence is conserved on average (Ramstead et al, 2023;Sakthivadivel, 2022b). This account of representation may thus be regarded as justifying (or dovetailing with) Quine's view that the categories in terms of which we experience the world are akin to theoretical posits, which earn their keep via their predictive and explanatory power with respect to sensation (Quine, 1960).…”

As without, so within: how the brain's temporo-spatial alignment to the environment shapes consciousness

“…In sum, the temporo-spatial alignment is closely related to the notion of symmetry in several ways. (i) Firstly, processes of temporal-spatial alignment result in symmetry between inner and outer states, (ii) with aligning processes understood as governed by informational-gauge-fields (Sengupta et al 2016;Fagerholm et al, 2020;Sakthivadivel, 2022) that preserve essential symmetries of brain and environment through their various sources of temporal and spatial coherence Safron, 2022).…”

Control flow in active inference systems

“…The Bayesian mechanics afforded by the FEP implies a (classical) thermodynamics; indeed, the FEP can be read as a constrained maximum entropy or caliber principle [71,72] (Sakthivadivel 2022, Sakthivadivel 2022). This follows from the fact that inference, i.e., self evidencing, entails belief updating and belief updating incurs a thermodynamic cost via the Jarzynski equality [73,74,75].…”