Paper:
Power-law efficient neural codes provide general link between perceptual bias and discriminability
Recent work in theoretical neuroscience has shown that efficient neural codes, which allocate neural resources to maximize the mutual information between stimuli and neural responses, give rise to a lawful relationship between perceptual bias and discriminability in psychophysical measurements (Wei & Stocker 2017, [1]). Here we generalize these results to show that the same law arises under a much larger family of optimal neural codes, which we call power-law efficient codes. These codes provide a unifying framework for understanding the relationship between perceptual bias and discriminability, and how it depends on the allocation of neural resources. Specifically, we show that the same lawful relationship between bias and discriminability arises whenever Fisher information is allocated proportional to any power of the prior distribution. This family includes neural codes that are optimal for minimizing L p error for any p, indicating that the lawful relationship observed in human psychophysical data does not require information-theoretically optimal neural codes. Furthermore, we derive the exact constant of proportionality governing the relationship between bias and discriminability for different choices of power law exponent q, which includes information-theoretic (q = 2) as well as "discrimax" (q = 1/2) neural codes, and different choices of decoder. As a bonus, our framework provides new insights into "anti-Bayesian" perceptual biases, in which percepts are biased away from the center of mass of the prior. We derive an explicit formula that clarifies precisely which combinations of neural encoder and decoder can give rise to such biases.

Review:
This paper generalizes previous work by Wei & Stocker (2017), by showing that a general class of variational objectives (“power-law efficient codes”) can produce the same psychophysical law that relates perceptual bias and discriminability. The work derives a strong form of the law, that holds in the asymptotically high SNR range (in which the prior is dominated by the likelihood), and a “weak” form which is nonlinear but works in a regime of non-negligible noise. In both cases, the paper explicitly derives the proportionality constants for the psychophysical law. The framework of this manuscript is slightly different than that used originally by Wei & Stocker (2017), but the authors show (mostly in the Supplementary Material) that the results hold also for the original framework (in particular, the encoding differs between the two, even though the two have the same Fisher information).  Comments:  This paper is technically sound, very well written and clear despite the complexity of the subject. The paper takes off from previous work by Wei and Stocker (2017) which proposed a novel psychophysical law (a lawful relation between perceptual bias and discriminability), but provides substantial novel content in a derivation of an entire class of “power-law” codes that would induce the same relation. This is important as it implies that the proposed psychophysical law could emerge for many different kinds of codes, and not only for an information maximizing code as it was assumed in previous work.  At the moment, it’s not clear where the ansatz for Eq. 21 comes from (I can guess it emerged from visual inspection of Figure 5C, and trying out a few functional forms). The authors might want to refer to Fig 5 here. (In fact, there is no reference to Fig 5 in the main text.)  This is a strong submission with novel theoretical results that lead to experimental predictions for psychophysical experiments (e.g., in terms of the coefficient of proportionality between bias and discriminability), and novel ideas that elaborate on the concept of efficient coding.  Typos: line 39: disriminability → discriminability line 138: differ → differs line 151.5 (unnumbered equation): remove either log on the left or exp on the right Eq 19: missing a minus sign in the argument of the exp line 315.5 (SI): in the first line of the equation, right-hand side, p(x) is missing line 430: minimzation → minimization line 432: wherein → therein line 444: supercede → supersede  After authors' feedback: Nothing to add - well done.