Sparse coupling and Markov blankets: A comment on "How particular is the physics of the Free Energy Principle?" by Aguilera, Millidge, Tschantz and Buckley

TL;DR

This paper clarifies the relationship between sparse coupling and Markov blankets in stochastic systems, providing conditions under which Markov blankets arise, applicable to both linear and nonlinear cases, thus informing the understanding of systems modeled by the Free Energy Principle.

Contribution

It derives precise conditions linking sparse coupling to Markov blankets in Gaussian steady-states, extending analysis to both linear and nonlinear stochastic differential equations.

Findings

Markov blankets are not guaranteed by sparse coupling in general.

Derived conditions specify when sparse coupling leads to Markov blankets.

Results apply to both linear and nonlinear stochastic systems.

Abstract

In this commentary, we respond to a technical analysis of the Free Energy Principle (hereafter: FEP) presented in "How particular is the physics of the Free Energy Principle" by Aguilera et al. In the target article, the authors analyzed certain sparsely coupled stochastic differential equations whose non-equilibrium steady-state densities are claimed--in previous FEP literature--to have a Markov blanket. The authors demonstrate that in general, Markov blankets are not guaranteed to follow from sparse coupling. The current commentary explains the relationship between sparse coupling and Markov blankets in the case of Gaussian steady-state densities. We precisely derive conditions under which causal coupling leads--or does not lead--to Markov blankets. Importantly, our derivations hold for both linear and non-linear stochastic differential equations. This result may shed light on the sorts of systems which we expect to have Markov blankets. Future work should focus on verifying whether these sorts of constraints are satisfied in realistic models of sparsely coupled systems.