Bayes vs. Weber: how to break a law of psychophysics

A classic tenet of psychophysics due to Weber is that human perceptual judgments are more variable for larger magnitudes. The more recent Bayesian paradigm proposes that human perception results from an optimal statistical inference conducted on the basis of noisy internal signals. Both are supported by a wealth of empirical evidence. Do the two conflict, and if so, which best reproduces human behavior? In two preregistered experiments, we manipulate the prior distribution and the reward function in a numerosity-estimation task. When the large numerosities are more frequent, and when they are more rewarding, the Bayesian observer exhibits an ‘anti-Weber’ behavior, in which larger magnitudes results in less variable responses. Human subjects exhibit a similar pattern, thus breaking a long-standing law of psychophysics by showing the opposite behavior. This allows subjects to minimize the errors they make about the more frequent or the more rewarding magnitudes. Nevertheless, model fitting suggests that subjects’ responses are best captured by a model that features a logarithmic encoding, a proposal of Fechner often regarded as accounting for Weber’s law. We thus obtain an anti-Weber behavior together with a Fechner encoding. Our results suggest that Weber’s law may be primarily due to the skewness of natural priors.