[Paper Review] A technical critique of the free energy principle as presented in "Life as we know it" and related works

Martin Biehl, Felix A. Pollock+1 more|arXiv (Cornell University)|Jan 12, 2020

Statistical Mechanics and Entropy15 references10 citations

TL;DR

This paper critically examines the free energy principle (FEP) as presented in Friston (2013) and related works, identifying three formal errors in the derivation of Bayesian inference from Markov blankets. It demonstrates that the claimed equivalence between variational and ergodic conditional densities does not hold in general, undermining the core interpretation of self-evidencing systems as performing Bayesian inference.

ABSTRACT

We summarize the argument in Friston (2013, https://doi.org/10.1098/rsif.2013.0475) and highlight some technical errors. We also discuss how these errors affect the very similar Friston (2014, https://doi.org/10.1109/JPROC.2014.2306251) and, where appropriate, mention consequences for the newer proposals in Friston (2019, arXiv:1906.10184v1 ) and Parr et al. (2019, https://royalsocietypublishing.org/doi/full/10.1098/rsta.2019.0159). The errors call into question the purported interpretation that the internal coordinates of every system with a Markov blanket will appear to engage in Bayesian inference. In particular, in addition to highlighting the implicit restriction to linear models, we identify three formal errors in the main argument of Friston (2013): The first concerns the rewriting of the equations of motion of systems with Markov blankets which turns out not to be generally correct. We prove the non-equivalence with a counterexample that exhibits a Markov blanket but does not satisfy the rewritten equations. Our counterexample also invalidates the corresponding (but more general) rewritten equations in the more recent Friston (2019). The second error concerns the Free Energy Lemma itself, which we prove, by counterexample, to be wrong in general. The third is the claim that the Free Energy Lemma, when it does hold, implies equality of variational density and ergodic conditional density. The interpretation in terms of Bayesian inference hinges on this point, and we hence conclude that it is unjustified. Additionally, we highlight that the definitions of the Markov blanket in Friston (2013) and Parr et al. (2019) are not equivalent and that the assumptions in Parr et al. (2019) may be too strong to allow for meaningful interpretation.

Motivation & Objective

• To identify and correct technical flaws in the derivation of the free energy principle as presented in Friston (2013).
• To assess the validity of the Free Energy Lemma and its implications for Bayesian inference in systems with Markov blankets.
• To evaluate the consistency and generality of Markov blanket definitions across Friston (2013), Friston (2019), and Parr et al. (2019).
• To determine whether the claimed equivalence between variational and ergodic conditional densities is mathematically justified.

Proposed method

• Constructing a counterexample to the rewritten equations of motion for systems with Markov blankets, demonstrating their non-equivalence in general cases.
• Proving the Free Energy Lemma to be invalid in general by constructing a counterexample where the lemma fails.
• Analyzing the claimed equivalence between variational density and ergodic conditional density, showing it does not follow from the lemma.
• Comparing definitions of Markov blankets in Friston (2013) and Parr et al. (2019), revealing non-equivalence and potential over-constraining assumptions.
• Extending the critique to Friston (2019) and Parr et al. (2019), showing that the same errors persist in newer formulations.
• Using formal mathematical reasoning to demonstrate that the assumptions underlying the FEP do not support the interpretation of Bayesian inference in general systems.

Experimental results

Research questions

• RQ1 Does the rewriting of the equations of motion for systems with Markov blankets hold universally, or are there counterexamples where it fails?
• RQ2 Is the Free Energy Lemma valid in general, or are there cases where it does not hold?
• RQ3 Does the Free Energy Lemma imply equality between variational density and ergodic conditional density under general conditions?
• RQ4 Are the definitions of Markov blankets in Friston (2013) and Parr et al. (2019) equivalent, or do they differ in critical ways?
• RQ5 Do the formal errors in the FEP derivation undermine the claim that all systems with Markov blankets perform Bayesian inference?

Key findings

• A counterexample is constructed that exhibits a Markov blanket but does not satisfy the rewritten equations of motion, invalidating the general validity of this transformation.
• The Free Energy Lemma is proven to be incorrect in general, as demonstrated by a counterexample where the lemma's conditions are met but the conclusion fails.
• The claimed equivalence between variational density and ergodic conditional density does not follow from the lemma, undermining the Bayesian interpretation of the FEP.
• The definitions of Markov blankets in Friston (2013) and Parr et al. (2019) are not equivalent, with the latter imposing stronger assumptions that may restrict applicability.
• The errors identified in Friston (2013) also invalidate the corresponding equations in Friston (2019), casting doubt on the broader generality of the FEP framework.