Bayesian Efficient Coding

Abstract

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 brains 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-theoretic and non-information-theoretic objectives within a single formalism - 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 covtropy, a novel family of posterior-functional losses 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 Laughlins seminal data on contrast coding in the blowfly large monopolar cell, and find that the measured response nonlinearity is better explained by minimizing Lp reconstruction error with p = 1/2 than by infomax, overturning a forty-year-old interpretation. Bayesian efficient coding thus enlarges the family of normatively optimal codes and provides a more general framework for understanding the design principles of sensory systems. Author summarySensory neurons work under tight budgets. They represent a rich, noisy world with limited spikes, limited dynamic range, and limited energy. Two long-standing ideas address this problem. One, called efficient coding, asks why neurons encode signals in the ways they do: which signals to amplify, which to filter out. A second, the Bayesian brain hypothesis, asks how the brain turns those signals into a percept: how to combine sensory evidence with prior knowledge to form a best guess about the world. We develop a framework that asks both questions at once. An optimal sensory code in our framework is specified by four interlocking ingredients: a prior description of the world, a model of how neurons respond to it, a limit on the resources they can spend, and a rule for what counts as a good internal representation. Classical information-maximizing codes emerge as one special case, one corner of a much larger family of Bayesian efficient codes. Applying this framework to Laughlins 1981 blowfly experiment, long held up as the textbook example of information maximization, we find that the flys neurons are better explained by minimizing decoding error than by maximizing information.