Particular flows and attracting sets: A comment on "How particular is the physics of the Free Energy Principle?" by Aguilera, Millidge, Tschantz and Buckley

TL;DR

This paper analyzes the flow fields of linear diffusions with solenoidal coupling, showing that for certain conditions, the flow of particular states aligns with variational free energy gradients, supporting an interpretive inference perspective.

Contribution

It demonstrates that under specific solenoidal couplings, the flow of particular states in stochastic systems aligns with free energy gradients, extending the inference interpretation beyond conditional modes.

Findings

Flow of particular states can point along free energy gradients with certain couplings.

Supports inference interpretation of internal states at arbitrary points in state space.

Provides geometric analysis of flow fields in linear diffusions.

Abstract

In this commentary, I expand on the analysis of the recent article "How particular is the physics of the Free Energy Principle?" by Aguilera et al. by studying the flow fields of linear diffusions, and particularly the rotation of their attracting sets in the presence of different types of solenoidal coupling. This analysis sheds new light on previous claims made in the FEP literature (and contested in the target article) that the internal dynamics of stochastic systems can be cast performing a gradient flow on variational free energy, and thus endowed with an inferential interpretation, i.e., as if internal states are performing inference about states external to the system. I express general agreement with the target article's statement that the marginal flow of internal states does not point along variational free energy gradients evaluated at the most likely internal state (i.e., the conditional mode). However, in this commentary I focus on the flow of particular states (internal and blanket states) and their variational free energy gradients, and show that for a wide but restricted class of solenoidal couplings, the average flow of these systems point along variational free energy gradients. This licenses a different but perhaps stronger re-description of the flow of particular states as performing inference, which importantly holds at arbitrary points in state space, not just at the conditional modes.