Reviewer #1 (Public review):

Summary:

The "number sense" refers to an imprecise and noisy representation of number. Many researchers propose that the number sense confers a fixed (exogenous) subjective representation of number that adheres to scalar variability, whereby the variance of the representation of number is linear in the number.

This manuscript investigates whether the representation of number is fixed, as usually assumed in the literature, or whether it is endogenous. The two dimensions on which the authors investigate this endogeneity are the subject's prior beliefs about stimuli values and the task objective. Using two experimental tasks, the authors collect data that are shown to violate scalar variability and are instead consistent with a model of optimal encoding and decoding, where the encoding phase depends endogenously on prior and task objectives. I believe the paper asks a critically important question. The literature in cognitive science, psychology, and increasingly in economics, has provided growing empirical evidence of decision-making consistent with efficient coding. However, the precise model mechanics can differ substantially across studies. This point was made forcefully in a paper by Ma and Woodford (2020, Behavioral & Brain Sciences), who argue that different researchers make different assumptions about the objective function and resource constraints across efficient coding models, leading to a proliferation of different models with ad-hoc assumptions. Thus, the possibility that optimal coding depends endogenously on the prior and the objective of the task, opens the door to a more parsimonious framework in which assumptions of the model can be constrained by environmental features. Along these lines, one of the authors' conclusions is that the degree of variability in subjective responses increases sublinearly in the width of the prior. And importantly, the degree of this sublinearity differs across the two tasks, in a manner that is consistent with a unified efficient coding model.

Comments:

(1) Modeling and implementation of estimation task

The biggest concern I have with the paper is about the experimental implementation and theoretical account of the estimation task. The salient features of the experimental data (Figure 1C) are that the standard deviations of subjects' estimated quantities are hump-shaped in the true stimulus x and that the standard deviation, conditional on the true stimulus x, is increasing in prior width. The authors attribute these features to a Bayesian encoding and decoding model in which the internal representation of the quantity is noisy, and the degree of noise depends on the prior - as in models of efficient coding (Wei and Stocker 2015 Nature Neuro; Bhui and Gershman 2018 Psych Review; Hahn and Wei 2024 Nature Neuro).

The concern I have is about the final "step" in the model, where the authors assume there is an additional layer of motor noise in selecting the response. The authors posit that the subject's selection of the response is drawn from a Gaussian with a mean set to the optimally decoded estimate x*(r), and variance set to a free parameter sigma_0^2. However, the authors also assume that the Gaussian distribution is "truncated to the prior range." This truncation is a nontrivial assumption, and I believe that on its own, it can explain many features of the data.

To see this, assume that there is no noise in the internal representation of x, there is only motor noise. This corresponds to a special case of the authors' model in which υ is set to 0. The model then reduces to a simple account in which responses are drawn from a Gaussian distribution centered at the true value of x, but with asymmetric noise due to the truncation. I simulated such a model with sigma_0=7. The resulting standard deviations of responses for each value of x (based on 1000 draws for each value of x), across the three different priors, reproduce the salient patterns of the standard deviation in Figure 1C: i) within each condition, the standard deviation is hump-shaped and peaks at x=60 and ii) conditional on x, standard deviation increases in prior width. The takeaway is that this simple model with only truncated motor noise - and without any noisy or efficient coding of internal representations - provides an alternative channel through which the prior affects behavior.

Of course, this does not imply that subjects' coding is not described by the efficient encoding and decoding model posited by the authors. However, it does suggest an important alternative mechanism for the authors' theoretical results in the estimation task. Moreover, some of the quantitative conclusions about the differences in behavior with the discrimination task would be greatly affected by the assumption of truncated motor noise.

Turning to the experiment, a basic question is whether such a truncation was actually implemented in the design. That is, was the range of the slider bar set to the range of the prior? (The methods section states that the size on the screen of the slider was proportional to the prior width, but it was unclear whether the bounds of the slider bar changed with the prior). If the slider bar range did depend on the prior, then it becomes difficult to interpret the data. If not, then perhaps one can perform analyses to understand how much the motor noise is responsible for the dependence of the standard deviation on both x and the prior width. Indeed, the authors emphasize that their model is best fit at α=0.48, which would seem to imply that the best fitting value of υ is strictly positive. However, it would be important to clarify whether the estimation procedure allowed for υ=0, or whether this noise parameter was constrained to be positive (i.e., clarify whether the estimation assumed noisy and efficient coding of internal representations).

(2) Differences across tasks

A main takeaway from the paper is that optimal coding depends on the expected reward function in each task. This is the explanation for why the degree of sublinearity between standard deviation and prior width changes across the estimation and discrimination task. But besides the two different reward functions, there are also other differences across the two tasks. For example, the estimation task involves a single array of dots, whereas the discrimination task involves a pair of sequences of Arabic numerals. Related to the discussion above, in the estimation task the response scale is continuous whereas in the discrimination task, responses are binary. Is it possible that these other differences in the task could contribute to the observed different degrees of sublinearity? It is likely beyond the scope of the paper to incorporate these differences into the model, but such differences across the two tasks should be discussed as potential drivers of differences in observed behavior.

If it becomes too difficult to interpret the data from the estimation task due to the slider bar varying with the prior range, then which of the paper's conclusions would still follow when restricting the analysis to the discrimination task?

(3) Placement literature

One closely related experiment to the discrimination task in the current paper can be found in Frydman and Jin (2022 Quarterly Journal of Economics). Those authors also experimentally vary the width of a uniform prior in a discrimination task using Arabic numerals, in order to test principles of efficient coding. Consistent with the current findings, Frydman and Jin find that subjects exhibit greater precision when making judgments about numbers drawn from a narrower distribution. However, what the current manuscript does is it goes beyond Frydman and Jin by modeling and experimentally varying task objectives to understand and test the effects on optimal coding. This contribution should be highlighted and contrasted against the earlier experimental work of Frydman and Jin to better articulate the novelty of the current manuscript.

Figures and data

Estimation task: the scale of subjects’ imprecision increases sublinearly with the prior width.

Discrimination task: the scale of subjects’ imprecision increases with the prior width; the relation is sublinear, but different than in the estimation task.

Estimation task: model fitting supports the hypothesis α = 1/2, both with pooled and individual responses.

Discrimination task: empirical across-subjects distribution of scaled best-fitting standard-deviation parameter.

Discrimination task: model fitting supports the hypothesis α = 3/4.