Constrained representations of numerical magnitudes

Imprecise probabilistic inference from sequential data.

“…This framework is typically used to model the processing of information that is external to the observer, although Azeredo da Silveira et al (2020) used it in a model of imprecise recall from memory. A different cognitive constraint appears in some encoding–decoding models of perception, which posit a cost (or a bound) proportional to a measure of the encoding capacity of the perceptual system, resulting in imprecise representations (Ganguli & Simoncelli, 2010; Morais & Pillow, 2018; Prat-Carrabin & Woodford, 2022). In any case, we surmise that changing the presentation of the evidence (e.g., leaving on screen the sequence of past rings) would change the cognitive cost of paying attention to it and result in different behavioral patterns.…”

Endogenous Precision of the Number Sense

“…With a = 1, this quantity approximates the expected quadratic loss that subjects in the estimation task should minimize in order to maximize their reward. And with a = 2, minimizing this loss is approximately equivalent to maximizing the reward in the discrimination task 25 . (The squared prior, in the expression of L 2 [ I ], corresponds to the probability of the co-occurrence of two presented numerosities that are close to each other, which is the kind of event most likely to result in errors in discrimination.…”

“…Numerosity studies yield similar results, pointing to an imprecise ‘number sense’ in the brain 7–13 . If the imprecision in representations reflects an optimal allocation of limited cognitive resources, as suggested by efficient-coding models 14–26 , then it should depend on the context in which representations are elicited 25,27 . Through an estimation task and a discrimination task, both involving numerosities, we show that the scale of subjects’ imprecision increases, but sublinearly, with the width of the prior distribution from which numbers are sampled.…”

“…First, in many studies, the encoding is assumed to optimize the mutual information between the external stimulus and the internal representations 19–21,23 , but it is seldom the case that this is actually the objective that an observer needs to optimize. An alternative possibility is that the encoding optimizes the observer’s current objective, which may vary depending on the task at hand 25,27 . Second, the nature of the resource that constrains the encoding is also unclear, and several possible limiting quantities are suggested in the literature (e.g., the expected spike rate, the number of neurons 17,18,21 , or a functional on the Fisher information, a statistical measure of the encoding precision 19,20,22,24,25 ).…”

“…An alternative possibility is that the encoding optimizes the observer’s current objective, which may vary depending on the task at hand 25,27 . Second, the nature of the resource that constrains the encoding is also unclear, and several possible limiting quantities are suggested in the literature (e.g., the expected spike rate, the number of neurons 17,18,21 , or a functional on the Fisher information, a statistical measure of the encoding precision 19,20,22,24,25 ). Third, most studies posit that the resource in question is costless, up to a certain bound beyond which the resource becomes depleted.…”