Figures and data in Endogenous precision of the number sense

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9 figures, 2 tables and 1 additional file

Figures

Figure 1

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

(a) Illustration of the estimation task: in each trial, a cloud of dots is presented on screen for 500 ms. Subjects are then asked to provide their best estimate of the number of dots shown. (b) Uniform prior distributions (from which the numbers of dots are sampled) in the three conditions of the task. (c) Standard deviation of the responses of the subjects (solid lines) and of the best-fitting model (dotted lines), as a function of the number of presented dots, in the three conditions. For each prior, five bins of approximately equal sizes are defined; subjects’ responses to the numbers falling in each bin are pooled together (thick lines) or not (thin lines). Orange crosses/circles: predictions for the Wide condition from the Narrow and Medium data, assuming an affine scaling of variance with squared width (crosses) or with width (circles). (d) Variance of subjects’ responses, as a function of the width of the prior (purple line) and of the squared width (gray line). Both lines show the same data; only the x-axis scale has been changed. (e) Subjects’ coefficients of variations, defined as the ratio of the standard deviation of estimates over the mean estimate, as a function of the presented number, in the three conditions. (f) Absolute error (solid line), defined as the absolute difference between a subject’s estimate and the correct number, and relative error (dashed line), defined as the ratio of the absolute error to the prior width, as a function of the prior width. In panels c-d, the responses of all the subjects (n=36) are pooled together; error bars show twice the standard errors.

Figure 2

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

(a) Illustration of the discrimination task: In each trial, subjects are shown five blue numbers and five red numbers, alternating in color, each for 500ms, after which they are asked to choose the color whose numbers have the higher average. (b) Uniform prior distributions (from which the numbers are sampled) in the two conditions of the task. (c) Proportion of choices ‘red’ in the responses of the subjects (solid lines) and of the best-fitting model (dotted lines), as a function of the difference between the two averages, in the two conditions. (d) Proportion of correct choices in subjects’ responses as a function of the absolute difference between the two averages divided by the square root of the prior width (left), by the prior width raised to the power $3 / 4$ (middle), and by the prior width (right). The three subpanels are different representations of the same data. In panels c and d, the responses of all the subjects are pooled together; error bars show the 95% confidence intervals (Narrow: n=31, Wide: n=32).

Figure 3

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Sublinear scaling of imprecision in Frydman and Jin’s risky-choice task.

In choices between a certain amount C and a lottery offering a probability $p = \frac{1}{2}$ of an amount X (and zero otherwise), participants’ proportion of choices for the lottery, as a function of $p X - C - b i a s$ , where $b i a s = .295$ is chosen so that the proportion is close to 50% at zero; and as a function of the width of the uniform prior distribution; with the abscissa unnormalized (first panel), or normalized by the prior width raised to the exponent 1/2, 3/4, or 1 (second, third, and fourth panels). The responses of all the subjects are pooled together; error bars show the 95% confidence intervals. The choice behavior of participants in this risky-choice task is consistent with that in our discrimination task (compare with Figure 2), and further supports our endogenous-precision model.

Figure 4

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Estimation task: comparison of models with and without cognitive noise.

(a) Standard deviation of responses as a function of the presented number, for the subjects (dotted lines), the best-fitting model with logarithmic encoding (solid lines), and with linear encoding (dashed lines). (b) Same as a, but with the model with only motor noise (constant across conditions) and no cognitive noise (solid lines). This model does not capture the variability of subjects as well as the models with cognitive noise in a. (c) Variance of responses as a function of the prior width, for the subjects (gray line), the best-fitting model with logarithmic encoding (solid pink line), and with linear encoding (dashed pink line), the model with only motor noise (constant across conditions) and no cognitive noise (solid green line), and the model with no cognitive noise and different degrees of motor noise in each condition (dotted green line). (d) For the model with no cognitive noise and prior-wise degrees of motor noise, distribution across subjects of the noise parameter in the three conditions. The solid lines show the empirical cumulative distribution functions, while the shaded areas show the distributions smoothed with a Gaussian kernel.

Figure 5

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Estimation task: hierarchical model estimates of response variability.

Fixed effects of a statistical model that includes subject-specific random effects (see main text). *Left:* Posterior-mean estimates of the standard deviations of responses as a function of the presented number, in the three conditions. *Right:* Posterior-mean estimates of the average variance as a function of the prior width. The results are similar to those presented in Figure 1. Shaded areas (*left*) and error bars (*right*) show the 5th and 95th percentile of the posteriors.

Figure 6

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Estimation task: hierarchical model estimates of mean responses and bias.

Fixed effects of a statistical model that includes subject-specific random effects (see main text). Posterior-mean estimates of the response (*left*) and the bias (*right*) as a function of the presented number, in the three conditions. Shaded areas show the 5th and 95th percentile of the posteriors. The dotted lines show the predictions of the best-fitting model ( $α = 1 / 2$ ) with logarithmic encoding.

Figure 7

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

The first panel shows the empirical cumulative distribution function (CDF) of the fitted parameter $\tilde{ν}$ , unscaled. The second, third, and fourth panels show the empirical CDF of $\tilde{ν}$ divided by $w^{α}$ , with $α = 1 / 2$ , $3 / 4$ , and 1, respectively.

Figure 8

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Discriminability for small vs. large numbers.

Subjects’ proportion of correct choices as a function of the absolute difference between the two averages, in the two conditions, for the trials in which both averages are below the middle value ( $x_{R}, x_{B} < 50$ ; dashed lines), and for those in which both are above ( $x_{R}, x_{B} > 50$ ; solid line). Abscissa bin widths: 3 in the Narrow condition and 8 in the Wide condition, except for the first bin whose width is half this value. Error bars indicate the 95% confidence interval. *: p < 0.05.

Figure 9

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Subjects’ behavior is stable in the two tasks.

Behavior of subjects in the estimation task (**a,b**) and in the discrimination task (**c,d**), in the first half of trials (dashed lines) and in the second half (solid lines). (a) Standard deviation of estimates as a function of the presented number (as in Figure 1c). (b) Variance of estimates as a function of the prior width and of the squared width (as in Figure 1d). (c) Choice probability as a function of the difference between the red and blue averages (as in Figure 2c). (d) Choice probability as a function of the absolute average difference divided by the prior width raised to the exponent 3/4 (as in Figure 2d, middle panel).

Tables

Table 1

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

Number of parameters (third-to-last column) and BICs of the Gaussian-representation model with the linear (second-to-last column) and the logarithmic encoding (last column) under different specifications regarding whether all subjects share the same values of the three parameters $α$ , $ν$ , and $σ_{0}$ (first three columns). ‘Shared’ indicates that the responses of all the subjects are modeled with the same value of the parameter. ‘Indiv.’ indicates that different values of the parameter are allowed for different subjects. ‘(x3)’ in the $σ_{0}$ column indicates that the parameter is different in each of the three conditions. For the parameter $α$ , ‘Fixed’ indicates that the value of $α$ is fixed (thus it is not a free parameter); when the parameter $α$ is ‘Shared’, it is a free parameter, and we indicate its best-fitting value in parentheses ( $α_{l i n}$ and $α_{l o g}$ for the linear and logarithmic encodings). ‘0’ in the $ν$ column indicates a model with no internal noise. The linear/logarithmic difference is thus meaningless for these models, and they have no parameter $α$ . In the first six rows of the table, all parameters are shared across the subjects, while in the remaining rows, at least one parameter is individually fit. In both cases, the lowest BIC (indicated by a star) is obtained for a model with a fixed parameter $α = 1 / 2$ .

$α$	$ν$	$σ_{0}$	Num. param.	BIC (lin)	BIC (log)
Fixed $α = 1$	Shared	Shared	2	81762.79	81443.44
Fixed $α = 1 / 2$	Shared	Shared	2	*81519.07	*81137.14
Shared $(α_{l i n} = .48, α_{l o g} = .44)$	Shared	Shared	3	81527.78	81141.34
Fixed $α = 0$	Shared	Shared	2	81864.77	81453.56
-	0	Shared	1	82679.89
-	0	Shared (x3)	3	82117.52
Fixed $α = 1$	Indiv.	Shared	37	81729.64	81343.21
Fixed $α = 1$	Shared	Indiv.	37	81746.34	81436.72
Fixed $α = 1$	Indiv.	Indiv.	72	81657.67	81333.07
Fixed $α = 1 / 2$	Indiv.	Shared	37	81427.11	81026.03
Fixed $α = 1 / 2$	Shared	Indiv.	37	81467.71	81089.06
Fixed $α = 1 / 2$	Indiv.	Indiv.	72	*81346.37	*80954.94
Shared $(α_{l i n} = .43, α_{l o g} = .44)$	Indiv.	Shared	38	81437.93	81028.97
Shared $(α_{l i n} = .45, α_{l o g} = .42)$	Shared	Indiv.	38	81472.85	81083.12
Shared $(α_{l i n} = .44, α_{l o g} = .44)$	Indiv.	Indiv.	73	81350.90	80955.05
Fixed $α = 0$	Indiv.	Shared	37	81745.04	81322.87
Fixed $α = 0$	Shared	Indiv.	37	81780.57	81369.95
Fixed $α = 0$	Indiv.	Indiv.	72	81588.93	81239.68
Indiv.	Shared	Shared	38	81444.60	81038.30
Indiv.	Indiv.	Shared	73	81571.48	81171.12
Indiv.	Shared	Indiv.	73	81366.40	80967.25
Indiv.	Indiv.	Indiv.	108	81453.52	81082.61
-	0	Indiv.	36	82529.71
-	0	Indiv. (x3)	108	82173.18

Table 2

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

Number of parameters (third column), BIC of the linear encoding model (fourth column), and of the logarithmic encoding model (last column), under different specifications regarding the parameter $α$ (first column) and the absence or presence of lapses (second column). In the bottom four lines, the models feature lapses, while they do not in the top four lines. With both encodings, the lowest BIC (indicated with a star) is obtained with lapses and with the specification $α = 3 / 4$ .

$α$	Lapses	Num. param.	BIC (lin.)	BIC (log)
Fixed $α = 1$	No	1	11737.03	11749.96
Fixed $α = 3 / 4$	No	1	11721.22	11745.35
Fixed $α = 1 / 2$	No	1	11815.86	11849.23
Free $(α_{l i n} = .84, α_{l o g} = .86)$	No	2	11723.22	11742.95
Fixed $α = 1$	Yes	2	11635.59	11630.97
Fixed $α = 3 / 4$	Yes	2	*11617.24	*11615.15
Fixed $α = 1 / 2$	Yes	2	11661.35	11659.28
Free $(α_{l i n} = .80, α_{l o g} = .81)$	Yes	3	11625.14	11622.66

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MDAR checklist: https://cdn.elifesciences.org/articles/101277/elife-101277-mdarchecklist1-v1.pdf
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Arthur Prat-Carrabin
Michael Woodford

(2026)

Endogenous precision of the number sense

eLife 13:RP101277.

https://doi.org/10.7554/eLife.101277.4

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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.

Sublinear scaling of imprecision in Frydman and Jin’s risky-choice task.

Estimation task: comparison of models with and without cognitive noise.

Estimation task: hierarchical model estimates of response variability.

Estimation task: hierarchical model estimates of mean responses and bias.

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

Discriminability for small vs. large numbers.

Subjects’ behavior is stable in the two tasks.

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

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

MDAR checklist

Downloads (link to download the article as PDF)

Open citations (links to open the citations from this article in various online reference manager services)

Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)

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

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