Abstract
One of the foundations of quantitative perceptual psychology is Weber’s observation of the nineteenth century: for many sensory attributes, the amount of just-detectable stimulus perturbation (perceptual threshold) is proportional to stimulus intensity. Fechner proposed that Weber’s Law arises from a logarithmic internal representation of these quantities, which when differentiated gives rise to the observed perceptual sensitivity. In apparent contradiction to Fechner’s proposed logarithmic relationship, Stevens (1957) found that observers’ ratings of perceived stimulus intensity followed a power law, with the power taking on a wide range of values across different stimulus attributes. Attempts have been made to reconcile these two conflicting quantitative accounts of the relationship between perception and stimulus intensity, but the problem remains unresolved, and continues to impede our understanding of the representation and comparison of perceptual quantities. We propose a resolution of this quandary, by separating the effects of both mean and variance of an abstract internal representation of stimuli. We assume that a rating scale, such as that used by Stevens, reflects the mean internal representation of stimulus intensity, but is unaffected by its variability. On the other hand, discrimination thresholds (as captured by Weber’s Law) depend on both the mean and variability of that internal representation, a relationship captured by Fisher Information. Stevens' Power Law mapping can be made consistent with Weber's Law by assuming internal noise whose standard deviation scales according to the same power law. This implies that the variance of internal representations must grow as the square of their mean, a super-Poisson property that has been attributed to fluctuations of response gain in sensory neurons (Goris et al 2015, Lin et al 2015). Considering the effects of both mean and variance of the internal representation brings us one step closer to a consistent mechanistic understanding of the established psychophysical observations.