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 brain’s ability to integrate sensory and prior sources of information in order to perform Bayesian inference. However, there is as yet no comprehensive theory connecting these two theoretical frameworks. We bridge this gap by formalizing a Bayesian theory of efficient coding. We define 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 function, quantifying the desirability or undesirability of various posterior distributions. Classic efficient codes can be seen as a special case in which the loss function is the posterior entropy, leading to a code that maximizes mutual information, but alternate loss functions give solutions that differ dramatically from information-maximizing codes. In particular, we show that decorrelation of sensory inputs, which is optimal under classic efficient codes in low-noise settings, can be disadvantageous for loss functions that penalize large errors. Bayesian efficient coding therefore enlarges the family of normatively optimal codes and provides a more general framework for understanding the design principles of sensory systems. We examine Bayesian efficient codes for linear receptive fields and nonlinear input-output functions, and show that our theory invites reinterpretation of Laughlin’s seminal analysis of efficient coding in the blowfly visual system.

One of the primary goals of theoretical neuroscience is to understand the functional organization of neurons in the early sensory pathways and the principles governing them. Why do sensory neurons amplify some signals and filter out others? What can explain the particular configurations and types of neurons found in early sensory system? What general principles can explain the solutions evolution has selected for extracting signals from the sensory environment?

Two of the most influential theories for addressing these questions are the “efficient coding” hypothesis and the “Bayesian brain” hypothesis. The efficient coding hypothesis, introduced by Attneave and Barlow more than fifty years ago, uses the ideas from Shannon’s information theory to formulate a theory normatively optimal neural coding [1, 2]. The Bayesian brain hypothesis, on the other hand, focuses on the brain’s ability to perform Bayesian inference, and can be traced back to ideas from Helmholtz about optimal perceptual inference [3–7].

A substantial literature has sought to alter or expand the original efficient coding hypothesis [5, 8–18], and a large number of papers have considered optimal codes in the context of Bayesian inference [19–26], However, the two theories have never been formally connected within a single, comprehensive theoretical framework. Here we propose to fill this gap by formulating a general Bayesian theory of efficient coding that unites the two hypotheses. We begin by reviewing the key elements of each theory and then describe a framework for unifying them. Our approach involves combining a prior and model-based likelihood function with a neural resource constraint and a loss functional that quantifies what makes for a “good” posterior distribution. We show that classic efficient codes arise when we use information-theoretic quantities for these ingredients, but that a much larger family of Bayesian efficient codes can be constructed by allowing these ingredients to vary. We explore Bayesian efficient codes for several important cases of interest, namely linear receptive fields and nonlinear response functions. The latter case was examined in an influential paper by Laughlin that examined contrast coding in the blowfly large monopolar cells (LMCs) [27]; we reanalyze data from this paper and argue that LMC responses are in fact better described as minimizing the average square-root error than as maximizing mutual information.