Natural selection finds natural gradient

Active inference provides a biologically plausible process theory of brain function. It specifies neuronal dynamics for state-estimation in terms of a gradient descent on (variational) free energy -- a measure of the fit between an internal (generative) model and sensory observations. When formulated for discrete state-space generative models, the free energy gradient turns out to be a prediction error -- plausibly encoded in the average membrane potentials of neuronal populations. Conversely, the expected probability of a state can then be expressed in terms of firing rates. We establish a construct validity to this scheme -- by showing that it is consistent with current models of neuronal dynamics -- and face validity, as it is able to synthesize a wide range of biologically plausible electrophysiological responses. We then show that these neuronal dynamics approximate natural gradient descent, a well-known optimisation algorithm from information geometry that prescribes locally optimal belief updates. Lastly, numerical simulations suggest that both schemes perform equally well on average. The performance of belief updating is scored in terms of information length, a measure of the distance traveled in information space, which has a direct interpretation in terms of metabolic efficiency. These results show that active inference is consistent with state-of-the-art models of neuronal dynamics and coincides with the natural gradient. This suggests that natural selection, by selecting the phenotypes that optimise metabolic and computational efficiency, has implicitly approximated the steepest direction in information space; namely, natural gradient descent.