Bayesian efficient coding
4 affiliations
- 1Champalimaud Foundation;
- 2Dept. of Neurobiology & Behavior, Stony Brook University
- 3Princeton University
- 4Princeton Neuroscience Institute, Princeton University
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. 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 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 Lp 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. Author summary: Sensory 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 Laughlin’s 1981 blowfly experiment, long held up as the textbook example of information maximization, we find that the fly’s neurons are better explained by minimizing decoding error than by maximizing information.
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BibTeX
@article{park2026bayesian,
title = {{Bayesian efficient coding}},
author = {Il Memming Park and Jonathan W. Pillow},
journal = {PLOS Computational Biology},
year = {2026},
month = nov,
doi = {10.1371/journal.pcbi.1014894},
url = {https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1014894},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Bayesian efficient coding
AU - Il Memming Park
AU - Jonathan W. Pillow
PY - 2026
DA - 2026/11/04
JO - PLOS Computational Biology
DO - 10.1371/journal.pcbi.1014894
UR - https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1014894
AB - 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 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 Lp 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. Author summary: Sensory 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 Laughlin’s 1981 blowfly experiment, long held up as the textbook example of information maximization, we find that the fly’s neurons are better explained by minimizing decoding error than by maximizing information.
SN - 1553-7358
SN - 1553-734X
ER -