Bayesian efficient coding

The efficient coding hypothesis, which proposes that neurons are optimized to maximize information about the environment, has provided a guiding theoretical framework for sensory and systems neuroscience. More recently, a theory known as the Bayesian Brain hypothesis has focused on the brain's ability to integrate sensory and prior sources of information in order to perform Bayesian inference. Although pieces of a connection between these two hypotheses have appeared in prior work, a general formulation that treats the optimality criterion as an arbitrary functional of the posterior distribution -- and thereby admits both information-based and other objectives within a single framework -- has remained largely implicit. Here we make this formulation explicit, developing a Bayesian theory of efficient coding that defines Bayesian efficient codes in terms of four basic ingredients: (1) a stimulus prior distribution; (2) an encoding model; (3) a capacity constraint, specifying a neural resource limit; and (4) a loss functional, quantifying the desirability or undesirability of various posterior distributions. Classic efficient codes arise as the special case in which the loss functional is the posterior entropy, leading to a code that maximizes mutual information, but alternate loss functionals give solutions that differ dramatically from information-maximizing codes. Within this framework we introduce {\it covtropy}, a novel family of losses defined on posterior distributions and parameterized by a single exponent, and use it to show that decorrelation of sensory inputs -- optimal under classic efficient codes in low-noise settings -- can be disadvantageous for objectives that penalize large errors. We then reanalyze Laughlin's seminal data on contrast coding in the blowfly large monopolar cell, and find that the measured response nonlinearity is better explained by minimizing $L_p$ reconstruction error with $p = 1/2$ than by information maximization, overturning a forty-year-old interpretation. Bayesian efficient coding thus enlarges the family of codes that are optimal under different objectives and provides a more general framework for understanding the design principles of sensory systems.