Some free-energy puzzles resolved: response to Thornton

Chris Thornton [1Thornton C. Some puzzles relating to the free energy principle: comment on Friston.Trends Cogn. Sci. 2010; 14: 53-54Abstract Full Text Full Text PDF PubMed Scopus (7) Google Scholar] poses some simple but key questions about the free-energy principle reviewed in [2Friston K. The free-energy principle: a rough guide to the brain?.Trends Cogn. Sci. 2009; 13: 293-301Abstract Full Text Full Text PDF PubMed Scopus (1007) Google Scholar]. These puzzles have simple and clear answers: Puzzle: “A generative model of causal structure in the environment is [then] obtained, on which basis the agent is able to infer the ‘causes of sensory samples’ [ibid. p. 294]. What is unclear is how this mechanism would function where sensory samples are ambiguous” [1Thornton C. Some puzzles relating to the free energy principle: comment on Friston.Trends Cogn. Sci. 2010; 14: 53-54Abstract Full Text Full Text PDF PubMed Scopus (7) Google Scholar]. Answer: One of the main motivations for the free-energy principle is its appeal to [approximate] Bayesian inference where ambiguities are resolved by priors [3Stone J.V. et al.Where is the light? Bayesian perceptual priors for lighting direction.Proc. Biol. Sci. 2009; 276: 1797-1804Crossref PubMed Scopus (24) Google Scholar]. Priors are mandated by the (ill-posed) problems created by ambiguity and empirical priors are an integral part of hierarchical inference [2,Box 3]. This is not theoretical hand waving; in biophysics, the free-energy formulation is used routinely to solve difficult ill-posed inverse problems (e.g. [4Phillips C. et al.An empirical Bayesian solution to the source reconstruction problem in EEG.Neuroimage. 2005; 24: 997-1011Crossref PubMed Scopus (164) Google Scholar]). Puzzle: “On the face of it, no particular stand is taken on emergence of the structures that mediate minimization. But looking at the definition of free-energy, we find a significant role being played by the variable ϑ. It is values of this variable that encapsulate the brain's representation of ‘environmental causes”’ [1Thornton C. Some puzzles relating to the free energy principle: comment on Friston.Trends Cogn. Sci. 2010; 14: 53-54Abstract Full Text Full Text PDF PubMed Scopus (7) Google Scholar]. Answer: The representations are not environmental causes ϑ but the sufficient statistics μ of the brain's recognition density q(ϑ;μ); these include synaptic activity and efficacy [2Friston K. The free-energy principle: a rough guide to the brain?.Trends Cogn. Sci. 2009; 13: 293-301Abstract Full Text Full Text PDF PubMed Scopus (1007) Google Scholar]. The implicit optimization of neuronal connections (i.e. perceptual learning) leads to hierarchical brain structures (models) that recapitulate causal structure in the sensorium. This optimization process can ‘prune’ the form or structure of the model (cf., synaptic pruning [5Tessier, C.R. and Broadie, K. (2009) Activity-dependent modulation of neural circuit synaptic connectivity. Front. Mol. Neurosci. 2:8. doi:10.3389/neuro.02.008.2009Google Scholar]) and is used routinely in model optimization (e.g. automatic relevance determination [6Tipping M.E. Sparse Bayesian learning and the Relevance Vector Machine.J. Mach. Learning Res. 2001; 1: 211-244Google Scholar]). Furthermore, one could regard natural selection as optimizing the structural form of models at an evolutionary scale, through minimizing free-energy (where it is called free-fitness [7Sella G. Hirsh A.E. The application of statistical physics to evolutionary biology.Proc. Natl. Acad. Sci. U S A. 2005; 102: 9541-9546Crossref PubMed Scopus (270) Google Scholar]). In a statistical setting, free-energy bounds on model evidence are used routinely in Bayesian model selection (where the log model evidence is negative surprise, e.g. [8Daunizeau J. et al.Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models.Physica D. 2009; 238: 2089-2118Crossref PubMed Scopus (148) Google Scholar];) (Figure 1). Puzzle: “With the framework providing no principle for deciding the range of ϑ, the brain's representation of the conditional density is inevitably a ‘slightly mysterious construct”’ [1Thornton C. Some puzzles relating to the free energy principle: comment on Friston.Trends Cogn. Sci. 2010; 14: 53-54Abstract Full Text Full Text PDF PubMed Scopus (7) Google Scholar]. Answer: The range of ϑ (the values it can take) is specified by the form of the (generative) model and the priors it entails. For example, the equation in Box 2 [2Friston K. The free-energy principle: a rough guide to the brain?.Trends Cogn. Sci. 2009; 13: 293-301Abstract Full Text Full Text PDF PubMed Scopus (1007) Google Scholar] specifies the range of hidden states in the world x(i)⊂ϑ with the range of a function, for example a neuronal activation function. The ‘slightly mysterious’ aspect of the recognition density is not its form (nor the implicit range of causes that are represented) but the fact that it is induced by the brain's physical states (which encode the recognition density). Puzzle: “It is unclear how introduction of the ‘free-energy’ concept, specifically, adds explanatory content…it is minimization of surprise that is explanatorily salient” [1Thornton C. Some puzzles relating to the free energy principle: comment on Friston.Trends Cogn. Sci. 2010; 14: 53-54Abstract Full Text Full Text PDF PubMed Scopus (7) Google Scholar]. Answer: The explanatory advance furnished by free-energy is fundamental: it provides a means to minimize surprise. This is because surprise cannot be quantified by an agent, whereas free-energy can. Again, this is not abstract hand waving; the free-energy bound on surprise (or log-evidence for a model) plays an essential role in physics [9Weissbach F. et al.High-order variational perturbation theory for the free energy.Phys. Rev. E. 2002; 66: 036129https://doi.org/10.1103/PhysRevE.66.036129Crossref Scopus (17) Google Scholar], machine learning [10Frey B.J. Jojic N. A comparison of algorithms for inference and learning in probabilistic graphical models.IEEE Trans. Pattern Anal. Mach. Intell. 2005; 27: 1392-1416Crossref PubMed Scopus (126) Google Scholar] and statistics [11Friston K. et al.Variational free energy and the Laplace approximation.NeuroImage. 2007; 34: 220-234Crossref PubMed Scopus (565) Google Scholar] for this reason.