Do estimates of numerosity really adhere to Weber’s law? A reexamination of two case studies

Abstract

Both humans and nonhuman animals can exhibit sensitivity to the approximate number of items in a visual array or events in a sequence, and across various paradigms, uncertainty in numerosity judgments increases with the number estimated or produced. The pattern of increase is usually described as exhibiting approximate adherence to Weber’s law, such that uncertainty increases proportionally to the mean estimate, resulting in a constant coefficient of variation. Such a pattern has been proposed to be a signature characteristic of an innate “number sense.” We reexamine published behavioral data from two studies that have been cited as prototypical evidence of adherence to Weber’s law and observe that in both cases variability increases less than this account would predict, as indicated by a decreasing coefficient of variation with an increase in number. We also consider evidence from numerosity discrimination studies that show deviations from the constant coefficient of variation pattern. Though behavioral data can sometimes exhibit approximate adherence to Weber’s law, our findings suggest that such adherence is not a fixed characteristic of the mechanisms whereby humans and animals estimate numerosity. We suggest instead that the observed pattern of increase in variability with number depends on the circumstances of the task and stimuli, and reflects an adaptive ensemble of mechanisms composed to optimize performance under these circumstances.

Learning Numerosity Representations with Transformers: Number Generation Tasks and Out-of-Distribution Generalization

“…A similar pattern of errors is observed when the Transformer is trained using the Non-Uniform Dots dataset (bottom panels), although in this case the sampling uncertainty associated with larger numerosities increases, and the model sometimes generates images with a mismatch of up to three items. Overall, these results are well-aligned with the existing empirical literature on human behavior, which suggests that numerosity estimates are distributed around the target mean and variability tends to increase with numerosity [ 42 , 44 ], and that numerosity estimation can be altered by confounding non-numerical magnitudes [ 21 , 25 ].…”

Numerical discrimination in Drosophila melanogaster

“…A strict interpretation of Weber’s law, which is thought to govern the ANS, might suggest that when performing numerical discrimination flies, and likely other animals, predominantly use ratio. On the other hand, there is evidence that Weber’s law approximations may not always apply, and that humans and animals use various representations of numerical values that are combinatorially deployed depending on the characteristics of the task 60 . This is consistent with our observations that flies also consider continuous properties of the visual stimuli to discriminate among sets of visual objects.…”

Ongoing dynamic calibration produces unstable number estimates.

“…However, the subsequent mapping from these internal representations to verbal estimates is assumed to be noise-free, leading to a constant coefficient of variation in their estimates. In other words, the variability of their estimates scales with the magnitude of the estimates as a result of Weber noise in the underlying number representations (though see recent findings in Testolin and McClelland (2021)). However, in the previous section we describe evidence that the bilinear slope of individual estimates may wobble across many trials, causing participants' estimate calibrations to "drift" over time.…”

“…Given that an individual's slow drift in estimate calibration is best detected across many trials and a large range of estimates, previous studies may have lacked a sufficient number of trials at large magnitudes to detect such effects (see e.g., Frank et al (2008), Frank et al (2012), and Gallistel (1990), Gordon (2004)). More recent work by Testolin and McClelland (2021) has also called into question the notion of a stable CoV.…”