Endogenous precision of the number sense

In each trial of a numerosity estimation task, subjects are asked to provide their best estimate of the number of dots contained in an array of dots presented for 500 ms on a computer screen (Figure 1a). In all trials, the number of dots is randomly sampled from a uniform distribution, hereafter called ‘the prior’, but the width of the prior, $w$, is different in three experimental conditions. In the ‘Narrow’ condition, the range of the prior is [50, 70] (thus the width $w$ is 20); in the ‘Medium’ condition, the range is [40, 80] (thus $w=40$); and in the ‘Wide’ condition, the range is [30, 90] (thus $w=60$; Figure 1b). In all three conditions, the mean of the prior (which is the middle of the range) is 60. As an incentive, the subjects receive for each trial a financial reward which decreases linearly with the square of their estimation error. Each condition comprises 120 trials, and thus often the same number is presented multiple times, but in these cases, the subjects do not always provide the same estimates. We now examine this variability in subjects’ responses.

Figure 1

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Studies on numerosity estimation with similar stimuli sometimes report that the standard deviation of estimates increases proportionally to the estimated number. This property, dubbed ‘scalar variability’, has been seen as a signature of numerical-estimation tasks, and more generally, of the ‘number sense’ (Izard and Dehaene, 2008). However, looking at the standard deviation of estimates as a function of the presented number, we find that it is not well described by an increasing line. In the three conditions, the standard deviation seems to be maximal near the center of the range (60) and to slightly decrease for numbers closer to the boundaries of the prior (Figure 1c; we report the mean estimates in Methods). Dividing each prior range into five bins of similar sizes, we compute the variance of estimates in each bin (see Methods). In the three conditions, the variance in the middle (third) bin is greater than the variances in the fourth and fifth bins (which contain larger numbers). These differences are significant (p-values of Levene’s tests of equality of variances: third vs. fifth bin, largest p-v. across the three conditions: 5e-6; third vs. fourth bin, Narrow condition: 0.009, Medium condition: 1.2e-5) except between the third and fourth bin in the Wide condition (p-v.: 0.12). This substantiates the conclusion that the standard deviation of estimates is not an increasing linear function of the number. Moreover, a hallmark of scalar variability is that the ‘coefficient of variation’, defined as the ratio of the standard deviation of estimates to the mean estimate, is constant (Izard and Dehaene, 2008). We find that in our experiment, it is decreasing for most of the numbers, in the three conditions (Figure 1e); this is consistent with the results of Testolin and McClelland, 2021. We conclude that the scalar-variability property is not verified in our data.

In fact, the most striking feature of the variability of estimates is not how it depends on the number, but how it strongly depends on the width of the prior, $w$ (Figure 1c and d). For instance, with the numerosity 60, the standard deviation of subjects’ estimates is 4.2 in the Narrow condition, 6.8 in the Medium condition, and 8.4 in the Wide condition, although these estimates are all obtained after the presentations of the same number of dots (60). Testing for the equality of the variances of estimates across the three conditions, for each number contained in all three priors (i.e. all the numbers in the Narrow range), we find that the three variances are significantly different, for all the numbers (largest Levene’s test p-value, across the numbers: 1e-7, median: 2e-15). The same pattern is observed when conducting the analysis separately on the first and second halves of the trials, suggesting that this behavior is stable throughout the experimental sessions (see Methods).

The variability of estimates increases with the width of the prior. This suggests that the imprecision in the internal representation of a number is larger when a larger range of numbers needs to be represented. This would be the case if internal representations relied on a mapping of the range of numbers to a normalized, bounded internal scale, and the estimate of a number resulted from a noisy readout (or a noisy storage) on this scale, as in ‘range-normalization’ models (Padoa-Schioppa, 2009; Kobayashi et al., 2010; Cai and Padoa-Schioppa, 2012; Soltani et al., 2012; Rangel and Clithero, 2012; Louie and Glimcher, 2012). Consider, for instance, the representation of a number $x$, obtained through its normalization onto the unit range [0, 1], and then read with noise, as

where $x_{min}$ is the lowest value of the prior, and $\epsilon$ a centered normal random variable with variance ${\nu }^2$. Suppose that the estimate, $\hat{x}$, is obtained by rescaling the noisy representation back to the original range, that is $\hat{x}=x_{min}+rw$ (we make this assumption for the sake of simplicity, but the argument we develop here is equally relevant for the more elaborate, Bayesian model we present below). The scale of the noise, given by $\nu$, is constant in the normalized scale; thus, in the space of estimates, the noise scales with the prior width, $w$. If we allow, in addition to the noise in estimates, for some amount of independent motor noise of variance ${\sigma }_0^2$ in the responses actually chosen by the subject, we obtain a model in which the variance of responses is ${\sigma }_0^2+{\nu }^2{w}^2$, that is, an affine function of the square of the width of the prior.

With the numerosity 60, the variance of subjects’ estimates is ${4.2}^2=17.64$ in the Narrow condition ($w=20$), and ${6.8}^2=46.24$ in the Medium condition ($w=40$): given these two values, the affine relation just mentioned predicts that in the Wide condition ($w=60$) the variance should be ${9.7}^2=93.91$. We find instead that it is ${8.4}^2=70.56$, i.e., about 25% lower than predicted, suggesting a sublinear relation between the variance and the square of the prior width. Indeed, the variance of estimates does not seem to be an affine function of the square of the prior width (Figure 1d, grey line and grey abscissa). Our investigations reveal that instead, the variance is significantly better captured by an affine function of the width — and not of the squared width (Figure 1d, purple line and purple abscissa).

As an additional illustration of this result, for each of the five bins mentioned above and defined for the three priors, we compute the predicted variance of estimates in the Wide condition on the basis of the variances in the Narrow and Medium conditions, and resulting either from the hypothesis of an affine function of the squared width, ${\sigma }_0^2+{\nu }^2{w}^2$, or from the hypothesis of an affine function of the width, ${\sigma }_0^2+{\nu }^{2}w$. The variances predicted with the former hypothesis all overestimate the variances of subjects’ responses (Figure 1c, orange crosses), but the predictions of the latter hypothesis appear consistent with the behavioral data (Figure 1c, orange circles).

We further investigate how the imprecision in internal representations depends on the width of the prior through a behavioral model in which responses result from a stochastic encoding of the numerosity, followed by a Bayesian decoding step. Specifically, the presentation of a number $x$ results in an internal representation, $r$, drawn from a Gaussian distribution with mean $\mu (x)$, where $\mu$ is an increasing function, and whose standard deviation, $\nu w^{\alpha }$, is proportional to the prior width raised to the power $\alpha$. That is,

where $\nu$ is a positive parameter that determines the baseline degree of imprecision in the representation, and $\alpha$ is a non-negative exponent that governs the dependence of the imprecision on the width of the prior. The Fisher information of this representation is $I(x)=({\mu }^{\mathrm{\prime }}(x)/(\nu w^{\alpha }){)}^2$. In short, $\mu (x)$ controls how the encoding precision varies with the number, $x$, while the exponent $\alpha$ controls how it varies with the prior width, $w$. For the encoding function, $\mu (x)$, a similar numerosity-estimation study finds that about a third of subjects behave consistently with a linear encoding, while a majority is consistent with a logarithmic encoding (Prat-Carrabin and Gershman, 2025). Thus, we consider both cases: a linear encoding, with the identity function $\mu (x)=x$; and a logarithmic encoding, $\mu (x)=\log (x)$.

Finally, the observer derives, from the internal representation $r$, the mean of the Bayesian posterior over $x$, $x^{\ast }(r)\equiv \mathbb{E}[x|r]$. We note that this estimate minimizes the squared-error loss, and thus maximizes the expected reward in the task. The selection of a response includes an amount of motor noise: the response, $\hat{x}$, is drawn from a Gaussian distribution centered on the Bayesian estimate, $x^{\ast }(r)$, with variance ${\sigma }_0^2$, truncated to the prior range, and rounded to the nearest integer. This model has three parameters (${\sigma }_0$, $\nu$, and $\alpha$).

The likelihood of the model is maximized for $\alpha =0.48$ and $\alpha =0.44$ with the linear and logarithmic encodings, respectively. These values are close to $1/2$ (and less close to 1), suggesting that the standard deviation is approximately a linear function of $\sqrt{w}$ (and the variance a linear function of $w$). The nested models obtained by fixing $\alpha =1/2$ yield slightly poorer fits (which is expected for nested models), but the differences in log-likelihood are small and thus the Bayesian Information Criterion (BIC), a measure of fit that penalizes larger numbers of parameters (Schwarz, 1978), is lower (i.e. better) for the constrained models with $\alpha =1/2$ (by 8.70 and 4.20, respectively; all BICs are reported in Table 1 in Methods). This indicates that setting $\alpha =1/2$ provides a parsimonious fit to the data that is not significantly improved by allowing $\alpha$ to differ from $1/2$. A different specification, $\alpha =1$, corresponds in the linear-encoding case to a normalization model similar to the one described above, but here with a Bayesian decoding of the internal representation. The BIC of this model is higher (by 244 and 306, respectively) than that with $\alpha =1/2$, indicating a much worse fit to the data. (Throughout, we report the models’ BICs even if they have the same number of parameters, so as to compare the values of a single metric). We emphasize that this large difference in BIC implies that the hypothesis $\alpha =1$ can be confidently rejected, in favor of the hypothesis $\alpha =1/2$ (in informal terms, it is not the case that the gray line in Figure 1d, showing the variance vs. the squared width, only appears curved because of some sampling noise; in fact, it is indeed not a straight line; while it is substantially more probable that the purple one, showing the variance vs. the width, corresponds indeed to a straight line).

The standard deviation of representations thus seems to increase linearly with the square root of the prior width, $\sqrt{w}$. The positive dependence results in larger errors when the prior is wider (Figure 1f, solid line). But the sublinear relation implies that the subjects in fact make smaller relative errors (relative to the width of the prior), when the prior is wider. In the Narrow condition, the ratio of the average absolute error to the width of the prior, $\frac{|\hat{x}-x|}{w}$, is 19.7%, i.e., the size of errors is about one fifth of the prior width. This ratio decreases substantially, to 14.5% and 11.6% in the Medium and Wide conditions, respectively, i.e., the size of errors is about one ninth of the prior width in the Wide condition (Figure 1f, dashed line). In other words, while the size of the prior is multiplied by 3, the relative size of errors is multiplied by $\frac{5}{9}\simeq 0.56$, and thus the absolute size of errors is multiplied by $3\cdot \frac{5}{9}\simeq 1.67$. If subjects had the same relative sizes of errors in both the Narrow and the Wide conditions, their absolute error would be multiplied by 3; conversely, the absolute error would be the same in the two conditions if the relative error was divided by 3. The behavior of subjects falls in between these two scenarios: they adopt smaller relative errors in the Wide condition, although not so much so as to reach the same absolute error as in the Narrow condition. Below, we show how this behavior is accounted for by a tradeoff between the performance in the task and a resource cost on the activity of the mobilized neurons.

The models we have considered combine a ‘cognitive’, encoding noise, parameterized by $\mu (.)$, $\nu$, and $\alpha$, and a ‘motor noise’ parameterized by ${\sigma }_0$. We have chosen the motor noise to be truncated to the prior range, as in our experiment the bounds of the slider correspond to the bounds of the prior (i.e. subjects cannot select a number that is not in the support of the prior). This naturally decreases the variability of responses near the bounds (Hahn and Wei, 2024; Figure 1c). With more numbers further from the bounds, the Wide prior is less subject to this effect than the Narrow prior, raising the possibility that our results (specifically, the increase of the variability with the prior width) originate in the truncated motor noise only, and not in the scaling of the cognitive noise. We have conducted a series of analyses to examine this possibility. We present them in detail in Methods; here we summarize these investigations and our conclusions. In short, we consider models that do not posit any cognitive noise (i.e. $\nu =0$), or in which there is some cognitive noise, but one that is insensitive to the prior range (i.e. $\nu \ne 0$ but $\alpha =0$). These models fit significantly worse than our best-fitting model (with $\nu \ne 0$ and $\alpha =1/2$). Moreover, these models fail to capture the variability in subjects’ responses, and the way it varies with the prior range. Furthermore, in a similar numerosity-estimation study with varying priors, but in which the slider is kept identical across all conditions, subjects also exhibit a linear increase of their response variance as a function of the prior range, as in our study, suggesting that this effect is not driven by the bounded slider range (Prat-Carrabin et al., 2026b). Finally, an fMRI study shows that numerosity-sensitive neural populations in human parietal cortex adapt their precision to the prior range, in a way that correlates with behavior, supporting the notion that the scaling in subjects’ variability originates in the scaling of their internal, cognitive noise (Prat-Carrabin et al., 2026a). Overall, we conclude that truncated motor noise does not account for our experimental findings, and that the data support the hypothesis of scaling cognitive noise.

Finally, here we have focused on the dependence of the imprecision on the prior width, $w$, as it is the main object of our study. The imprecision may also depend on the numerosity, $x$, and indeed we find that the logarithmic encoding yields a better fit than the linear one, implying that larger numerosities are encoded with lower precision. We discuss further below this ‘compression’ of the number line; here we note that regardless of the choice of encoding, we reach the same conclusions regarding the sublinear relation between the imprecision of representations and the width of the prior. We now ask whether subjects exhibit, in a discrimination task, the same sublinear scaling.

In many decision situations, instead of providing an estimate, one is required to select the better of two options. We thus investigate experimentally the behavior of subjects in a discrimination task. In each trial, subjects are presented with two interleaved series of numbers, five red and five blue numbers, after which they are asked to choose the series that had the higher average (Figure 2a). Each number is shown for 500 ms. Two experimental conditions differ by the width of the uniform prior from which the numbers (both blue and red) are sampled: in the Narrow condition, the range of the prior is [35, 65] (the width of the prior is thus $w=30$) and in the Wide condition, the range is [10, 90] (the width is thus $w=80$; Figure 2b). After each decision, subjects receive a number of points equal to the average that they chose. At the end of the experiment, the total sum of their points is converted to a financial reward (through an increasing affine function).

Figure 2

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Subjects in this experiment sometimes make incorrect choices (i.e. they choose the color whose numbers had the lower average), but they make less incorrect choices when the difference between the two averages is larger, and the proportion of trials in which they choose ‘red’ is a sigmoid function of the difference between the average of the red numbers, $x_R$, and the average of the blue numbers, $x_B$ (Figure 2c). In the Narrow condition, this proportion reaches 60% when the difference in the averages is 1, and 90% when the difference is 7. In the Wide condition, we find that the slope of this psychometric curve is less steep: subjects reach the same two proportions for differences of about 2.4 and 12.6, respectively. The separate analyses of the first and second halves of the trials suggest that this pattern is stable over the course of the experiment (see Methods).

In the Wide condition, it thus requires a larger difference between the red and blue averages for the subjects to reach the same discrimination threshold; put another way, the same difference in the averages results in more incorrect choices in the Wide condition than in the Narrow condition. As with the estimation task, this suggests that the degree of imprecision in representations is larger when the range of numbers that must be represented is larger. To estimate this quantitatively, we turn to the predictions of the model presented above, here considered in the context of the discrimination task: in this model, the average $x_C$, where C is ‘blue’ or ‘red’ (denoted by B and R, respectively), results in an internal representation, $r_C$, drawn from a Gaussian distribution with mean $\mu (x_C)$, where ${\mu }^{\mathrm{\prime }}>0$, and whose variance, ${\nu }^2{w}^{2\alpha }$, is proportional to the prior width raised to the exponent $2\alpha$, i.e., $r_C|x_C\sim N(\mu (x_C),{\nu }^2{w}^{2\alpha })$. Given the (independent) representations $r_B$ and $r_R$, the subject, optimally, compares the Bayesian estimates for each quantity, $x^{\ast }(r_B)$ and $x^{\ast }(r_R)$, and chooses the greater one. As the Bayesian estimate is an increasing function of the representation, the probability that the subject chooses ‘red’, conditional on two averages $x_B$ and $x_R$, is the probability that $r_R$ be larger than $r_B$, that is

where $\mathrm{\Phi }$ is the cumulative distribution function of the standard normal distribution. Here also we consider both a linear encoding ($\mu (x)=x$) and a logarithmic encoding ($\mu (x)=\log (x)$). In this task, the model fitting does not strongly favor one over the other, thus we focus here on the linear encoding, and we discuss the logarithmic encoding further below.

The choice probability (Equation 3) is predicted to be a function of the ratio $\frac{x_R-x_B}{w^{\alpha }}$ of the difference between the two averages over the width of the prior raised to the power $\alpha$, and therefore the same choice probability should be obtained across conditions as long as this ratio is the same. In Figure 2d, we show for different values of $\alpha$ the subjects’ proportions of correct responses as a function of the absolute value of this ratio, so as to be able to examine closely the difference between the resulting choice curves in the two conditions. The case $\alpha =1$ corresponds, as above, to the hypothesis that the standard deviation of internal representations is a linear function of the width, $w$, that is, a normalization of the numbers by the width of the prior. But we find that the proportion of correct choices as a function of the ratio $|x_R-x_B|/w$ is greater in the Wide condition than in the Narrow condition (Figure 2d, last panel). In other words, in the Wide condition, the subjects are more sensitive to the normalized difference than in the Narrow condition. This suggests that between the Narrow and the Wide conditions, the imprecision in representations does not change in the same proportions as does the prior width; specifically, it suggests a sublinear relation between the scale of the imprecision and the width of the prior.

As seen in the previous section, the behavioral data in the estimation task precisely suggest such a sublinear relation and more precisely point to the exponent $\alpha =1/2$, i.e., to a linear relation between the standard deviation and the square root of the width, $\sqrt{w}$. But the proportion of correct choices as a function of the corresponding ratio, $|x_R-x_B|/\sqrt{w}$, is greater in the Narrow condition than in the Wide condition (Figure 2d, first panel). The sublinear relation, thus, is not the same in the two tasks, and the data suggest in the case of the discrimination task an exponent $\alpha$ greater than $1/2$, but lower than 1. Indeed, we find that the choice curves in the two conditions match very well with $\alpha =3/4$ (Figure 2d, middle panel).

Model fitting substantiates this result. We add to our model (in which the probability of choosing ‘red’ is given by Equation 3) the possibility of ‘lapse’ events, in which either response is chosen with probability 50%; an additional parameter, $\eta$, governs the probability of lapses. (We reach the same conclusions with a model with no lapse, but this model with lapses yields a better fit; see Table 2 in Methods.) With the linear (logarithmic) encoding, the BIC of this model with $\alpha =3/4$ is lower (i.e., better) by 44.11 (44.13) than that with $\alpha =1/2$, and by 18.3 (15.8) than that with $\alpha =1$, indicating strong evidence rejecting the hypotheses $\alpha =1/2$ and $\alpha =1$, in favor instead of the hypothesis of an exponent $\alpha$ equal to $3/4$. Notwithstanding the theoretical reasons, presented below, that motivate our focus on this specific value of the exponent in addition to the good fit to the data, we can let $\alpha$ be a free parameter, in which case its best-fitting value is 0.80 (0.81), and thus close to $3/4$. This model’s BIC is, however, higher (i.e., worse) by 7.9 (7.5) than that of the model with $\alpha$ fixed at $3/4$, which indicates strong evidence (Kass and Raftery, 1995) in favor of the equality $\alpha =3/4$. In sum, our best-fitting model is one in which the standard deviation of the internal representations is a linear function of the prior width raised to the power 3/4. As with the estimation task, this sublinear relation implies that subjects are relatively more precise when the prior is wider. This allows them to achieve a significantly better performance in the Wide condition than in the Narrow condition (with 80.2% and 77.4% of correct responses, respectively; p-value of Fisher’s exact test of equality of the proportions: 9.5e-5).

The subjects’ behavioral patterns in the estimation task and in the discrimination task suggest that the scale of the imprecision in their internal representations increases sublinearly with the range of numerosities used in a given experimental condition. Specifically, the scale of the imprecision seems to be a linear function of the prior width raised to the power $1/2$, in the estimation task, and raised to the power $3/4$, in the discrimination task. We now show that these two exponents, $1/2$ and $3/4$, arise naturally if one assumes that the observer optimizes the expected reward in each task, while incurring a cost on the activity of the neurons that encode the numerosities.

Inspired by models of perception in neuroscience (Grzywacz and Balboa, 2002; Stocker and Simoncelli, 2006b; Stocker and Simoncelli, 2006a; Ganguli and Simoncelli, 2010; Ganguli and Simoncelli, 2014; Wei and Stocker, 2015; Ganguli and Simoncelli, 2016; Wang et al., 2016; Park and Pillow, 2017; Morais and Pillow, 2018; Prat-Carrabin and Woodford, 2021; Zhang and Stocker, 2022), we consider a two-stage, encoding-decoding model of an observer’s numerosity representation. In the encoding stage, a numerosity $x$ elicits in the brain of the observer an imprecise, stochastic representation, $r$, while the decoding stage yields the mean of the Bayesian posterior, which is the optimal decoder in both tasks. The model of Gaussian representations that we use throughout the text is one example of such an encoding-decoding model.

The encoding mechanism is characterized by its Fisher information, $I(x)$, which reflects the sensitivity of the representation’s probability distribution to changes in the stimulus $x$. The inverse of the square root of the Fisher information, $1/\sqrt{I(x)}$, can be understood as the scale of the imprecision of the representation about a numerosity $x$. More precisely, it is approximately — when $I(x)$ is large — the standard deviation of the Bayesian-mean estimate of $x$ derived from the encoded representation. (For smaller $I(x)$, the standard deviation of the Bayesian-mean estimate increasingly depends on the shape of the prior; with a uniform prior, it decreases near the boundaries.) The variability in subjects’ responses in the estimation task, and their choice probabilities in the discrimination task, reported above, is thus an indirect measure of the Fisher information of their encoding process.

Moreover, the expected squared error of the Bayesian-mean estimate of $x$ is approximately the inverse of the Fisher information, $1/I(x)$. We thus consider the generalized loss function

where $\pi (x)$ is the prior distribution from which $x$ is sampled. 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 (Prat-Carrabin and Woodford, 2021). (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.)

In both cases, a more precise encoding, i.e., a greater Fisher information, results in a smaller loss. We assume, however, that constraints on the encoding prevent the observer from choosing an infinitely large Fisher information. Specifically, we assume that the encoding results from an accumulation of independent, identically distributed signals, but the precision of each signal is limited, and each of them entails a cost. Formally, we posit, first, that the Fisher information of one signal, $I_1(x)$, is subject to the constraint:

This constraint appears in many other efficient-coding models in the literature (Wei and Stocker, 2015; Wei and Stocker, 2016; Wang et al., 2016; Morais and Pillow, 2018; Prat-Carrabin and Woodford, 2021; Prat-Carrabin and Woodford, 2022), and it arises naturally for unidimensional encoding channels (Prat-Carrabin and Woodford, 2021; e.g., for a neuron with a sigmoidal tuning curve, it is equivalent to assuming that the range of possible firing rates is bounded). Second, we assume that the observer incurs a cost each time a signal is emitted (e.g., the energy resources consumed by action potentials; Laughlin et al., 1998; Hasenstaub et al., 2010; Sengupta et al., 2010). The total cost is thus proportional to the number of signals, which we denote by $n$. More signals, however, allow for a better precision: specifically, under the assumption of independent signals, the total Fisher information resulting from $n$ signals is the sum of the Fisher information of each signal, that is $I(x)=n{I}_1(x)$.

A trade-off ensues between the increased precision brought by accumulating more signals and the cost of these signals. We assume that the observer chooses the function $I_1(.)$ and the number $n$ of signals that solve the minimization problem

where $\lambda >0$. We can first solve this problem for the Fisher information of one signal, $I_1(x)$. In the case of a uniform prior of width $w$, we find that it is zero outside of the support of the prior, and

for any $x$ on the support of the prior. This intermediate result corresponds to the optimal Fisher information of an observer who is not allowed to choose the number of signals, $n$ (and who receives instead $n=1$ signal). It is the solution predicted by the efficient-coding models mentioned above, that include the constraint in Equation 5, but that do not allow for the observer to choose the amount of signals, $n$. With this solution, the scale of the observer’s imprecision, $1/\sqrt{I_1(x)}$, is proportional to $w$, and it does not depend on the task — contrary to our experimental results.

Solving the optimization problem (Equation 6) for $n$, in addition to $I_1(x)$, we find that with a uniform prior, the optimal number is proportional to $w$ in the estimation task, and to $\sqrt{w}$ in the discrimination task (specifically, treating $n$ as continuous, we obtain $n=w^{\frac{3-a}{2}}/\sqrt{\lambda K}$). In other words, the observer chooses to obtain more signals when the prior is wider, and in a way that depends on the task. We give the general solution for the total Fisher information, $I(x)=n{I}_1(x)$, in the case of a prior $\pi (x)$ that is not necessarily uniform:

where $\theta =\lambda /K$. This implies that the optimal Fisher information vanishes outside of the support of the prior; and in the case of a uniform prior of width $w$, $I(x)$ is constant, as

for any $x$ such that $\pi (x)\ne 0$.

The scale of the imprecision of internal representations, $1/\sqrt{I(x)}$, is thus predicted to be proportional to the prior width raised to the power $1/2$, in the estimation task, and raised to the power $3/4$, in the discrimination task. As shown above, we find indeed that in these tasks, the imprecision of representations not only increases with the prior width, but it does so in a way that is quantitatively consistent with these two exponents. As for the model of Gaussian representations that we have considered throughout the text, we have noted that its Fisher information is $({\mu }^{\mathrm{\prime }}(x)/(\nu w^{\alpha }){)}^2$. The linear version of this model (with $\mu (x)=x$) is thus consistent with the predictions of Equation 9, if $\alpha =1/2$ for the estimation task, and $\alpha =3/4$ for the discrimination task. These are precisely the two values that best fit the data.

Many efficient-coding models in the literature feature a different objective, the maximization of the mutual information (Wei and Stocker, 2015; Wei and Stocker, 2016; Ganguli and Simoncelli, 2016); but a single objective cannot explain our different findings in the two tasks (namely, the different dependence on the prior width). And many models, as mentioned, feature the same constraint (Equation 5, or a generalized one Wei and Stocker, 2015; Wei and Stocker, 2016; Wang et al., 2016; Morais and Pillow, 2018), but without the endogenous choice that we have assumed regarding the amount of signals. This predicts that the imprecision scales proportionally to the prior width, regardless of the objective of the task. This hypothesis thus cannot account either for the difference that we find between the two tasks, nor for the observed sublinear scalings. By contrast, we assume that it is the task’s expected reward that is maximized, and that the amount of utilized encoding resources can be endogenously determined: our model is thus able to predict not only that the behavior should depend on the prior, but also that this dependence should change with the task; and it makes quantitative predictions that coincide with our experimental findings.

An experiment that shares similarities with our discrimination task, but in the context of risky choice behavior, is the one conducted by Frydman and Jin, in which they manipulate across experimental conditions the uniform prior distributions of the lottery amounts that they present to participants (Frydman and Jin, 2021a). They find that participants are less precise when stimuli are drawn from a wider prior, and thus our results are qualitatively consistent with theirs. But the resulting dataset, made publicly available (Frydman and Jin, 2021b), enables us to test the predictions of our theory against human choices collected in a rather different experimental paradigm. We thus look at the participants’ probability of choosing a risky lottery instead of a certain amount, as a function of the difference between the lottery’s expected value and the certain amount (we also add a small bias term to the certain option; such bias is not necessary with our discrimination data, presumably because of the inherent symmetry of our task). We provide more detail on our analysis of this dataset in Methods.

We find, as the authors did, and similarly to our discrimination task, that the participants are more precise when the proposed amounts are sampled from a Narrow prior, in comparison to a Wide prior (Figure 3, first panel). But we also find, as in our discrimination task, that when normalizing the value difference by the prior width, participants are more sensitive to this normalized difference in the Wide condition than in the Narrow one, suggesting that their imprecision scales across conditions by a smaller factor than the prior width (Figure 3, last panel). And we find, consistent with our discrimination data and with our theory, that choice probabilities in the two conditions match very well when normalizing the difference by the prior width raised to the exponent 3/4 (Figure 3, third panel). Model fitting supports this observation. We fit the data to the model described by Equation 3 (where $x_R$ and $x_B$ are replaced by the lottery’s expected value and by the certain amount), with the addition of a lapse probability and of a bias, and with different values of the exponent $\alpha$. The best-fitting model is the one with $\alpha =3/4$. Its BIC (35,419) is lower than those of the models with $\alpha =1$, $1/2$, and 0 (by 142, 39, and 514, respectively). It is also lower by 2.14 than a model in which $\alpha$ is left as a free parameter (in which case the best-fitting $\alpha$ is 0.68, a value not far from 3/4). We emphasize that these BIC values indicate that the hypotheses $\alpha =0$ and $\alpha =1$ are clearly rejected, i.e., the participants’ imprecision increases with the prior width ($\alpha >0$), but sublinearly ($\alpha <1$). In other words, the responses collected by Frydman and Jin in a risky-choice task are quantitatively consistent with our results obtained in a number-discrimination task, and they further substantiate our model of endogenous precision.

Figure 3

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We moreover note that their proposed model is similar to ours, in that the decision-maker is allowed to optimize a noisy encoding scheme to the prior, subject to a ‘capacity constraint’ on the number $n$ of encoding signals that can be obtained. Crucially, this capacity constraint is assumed to be a property of the decision-maker that does not change across priors, and thus $n$ is fixed across prior widths. Therefore, their model predicts that the participants’ imprecision should scale linearly with the prior width, as in Equation 7 above where the number of signals, $n$, was not optimized. We note that when they fit this parameter, $n$, separately across conditions, they find that it is larger with the wider prior. This is precisely what our model of endogenous precision predicts (as detailed above). In turn, this predicts a sublinear scaling of the imprecision, instead of the linear one that would result from a fixed $n$, and indeed we find a sublinear scaling in both their dataset and ours. What is more, in both datasets, the sublinear scaling is best captured by the exponent $\alpha =3/4$, as we predict.

Our model successfully accounts for the dependence of subjects’ imprecision on the prior width. It predicts no variation of the internal imprecision across numbers (the variance of estimates, however, may change near the bounds of the prior). Yet, a prominent idea in studies of the number sense is that the brain represents numbers on a logarithmic, compressed scale (Dehaene, 2011; Barretto-García et al., 2023; Khaw et al., 2021). Thus, for both tasks, we have fit a version of our Gaussian model that features a logarithmic encoding, $\mu (x)=\log (x)$. All our conclusions regarding the scaling of the imprecision remain the same regardless of this choice of the encoding. In the estimation task, with the best-fitting model ($\alpha =1/2$), the logarithmic encoding yields a significantly lower BIC than the linear one (by more than 380; see Table 1 in Methods). The results are less clear in the discrimination task, where the BIC with the logarithmic encoding (and $\alpha =3/4$) is lower by 2.1 when pooling together the responses of all the subjects, but it is larger by 2.6 when fitting each subject individually. (Looking directly at the proportion of correct responses, we find some evidence that subjects discriminate smaller numbers with a slightly higher success rate; we present this analysis in Methods.) We conduct a ‘Bayesian model selection’ procedure to estimate the relative prevalence of each encoding among subjects (Rigoux et al., 2014; see Methods). The resulting estimate of the fraction of the population that is best fit by the logarithmic encoding is 87.6% in the estimation task, and 45.9% in the discrimination task (vs. 12.4% and 54.1% for the linear encoding).

The large fraction of subjects best fit by the logarithmic encoding prompts us to consider the hypothesis that for some subjects, this logarithmic compression is an additional constraint on the possible encoding schemes. In the model above, the signal could take any form as long as its Fisher information verified the constraint in Equation 5. We now consider a strong, additional constraint: that over the support of the prior, the Fisher information of the signal must be of the form that one would obtain with a logarithmic encoding, that is $I_1(x)\propto 1/x^2$. (For the sake of generality, we choose this specification instead of directly assuming a logarithmic encoding, because other types of encoding schemes yield a Fisher information of this form, e.g. one with ‘multiplicative noise’ Zhou et al., 2024; we do not seek, here, to distinguish between these different possibilities). We solve the same optimization problem as above (Equation 6), but now with this additional constraint. We find that the resulting optimal Fisher information is approximately (see Methods):

for any $x$ on the support of the prior, and where $x_{\text{mid}}$ is the middle of the prior and $\theta$ is a constant. These Fisher-information functions differ from those in Equation 9 in that they fall off as $1/x^2$, consistent with our additional constraint. However, we note that the dependence on the prior width, $w$, is identical: here also, the imprecision $1/\sqrt{I(x)}$ is proportional to $\sqrt{w}$, in the estimation task, and to $w^{3/4}$, in the discrimination task.

In its logarithmic variant ($\mu (x)=\log (x)$), the Fisher information of the model of Gaussian representations that we have considered throughout is $1/(x\nu w^{\alpha }{)}^2$. It is thus consistent with the predictions just presented (Equation 10), if $\alpha =1/2$ for the estimation task, and $\alpha =3/4$ for the discrimination task, that is here also the two values that best fit the data. Overall, we conclude that with both linear and logarithmic encoding schemes, our efficient-coding model — wherein the degree of imprecision is endogenously determined — accounts for the task-dependent sublinear scaling of the imprecision that we observe in behavioral data.

We compare the responses of the subjects and of the Gaussian-representation model, with $\alpha =1/2$ in the estimation task and $\alpha =3/4$ in the discrimination task, and with the logarithmic encoding. In both cases, the parameter $\nu$ governs the imprecision in the internal representation, and a second parameter corresponds to additional response noise: the motor noise, parameterized by ${\sigma }_0^2$, in the estimation task, and the lapse probability, $x$, in the discrimination task. The behavior of the model, across the two tasks and the different priors, reproduces that of the subjects (Figures 1c and 2c, dotted lines). In the estimation task, the standard deviation of estimates increases as a function of the prior width, as it does in subjects’ responses. The Bayesian estimate, $x^{\ast }(r)$, depends on the prior, and its variability decreases for numerosities closer to the edges of the uniform prior. Hence the standard deviation of the model’s estimates adopts an inverted U-shape similar to that of the subjects (Figure 1c). In the discrimination task, the model’s choice-probability curve is steeper in the Narrow condition than in the Wide condition, and the two predicted curves are close to the subjects’ choice probabilities (Figure 2c). We emphasize that how the internal imprecision scales with the prior width is entirely determined by our theoretical predictions (Equation 9); these quantitative predictions allow our model to capture the subjects’ imprecise responses simultaneously across different priors.

In our analysis of the data collected by Frydman and Jin (Frydman and Jin, 2021a; Frydman and Jin, 2021b), we removed the responses of one subject, who chose the certain option on all trials in one of the conditions; the responses in all the trials in which the response time was less than 0.5 s; the responses of the second condition that each subject experienced; and the 30 ‘adaptation’ trials. This is consistent with what the authors did in their study.

For both the estimation and the discrimination task, we analyze separately the responses obtained in the trials in the first and second halves of each condition (Figure 9). In the estimation task, the standard deviations of responses, as a function of the presented number and of the prior width, are very similar in the two halves (Figure 9a). The Bonferroni-Holm-corrected p-values of Levene’s tests of equality of the variances across the two halves are all above 0.13, and thus we do not reject the hypothesis that the variance in the first half of the trials is equal to the variance in the second half. Moreover, the variance in both halves appears to be a linear function of the width, rather than the squared width (Figure 9b). We conclude that the behavior of subjects in the estimation task is stable across each experimental condition, including the sublinear scaling of their imprecision.

Figure 9

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In the discrimination task, the subjects’ choice probabilities, as a function of the difference between the averages of the red and blue numbers, are similar in the first and second halves of trials (Figure 9c). The Bonferroni-Holm-corrected p-values of Fisher exact tests of equality of proportions (in bins of the average difference that contain about 500 trials each) are all above 0.9, and thus we do not reject the hypothesis that the choice probabilities are equal, in the first and second halves of the trials. Furthermore, the choice probabilities as a function of the absolute average difference normalized by the prior width raised to the exponent $3/4$ are all similar, across session halves and across prior widths, suggesting that the sublinear scaling that we find is a stable behavior of subjects (Figure 9d).

Overall, we conclude that the behavior we exhibit in both tasks is stable over the course of each experimental condition. We note that in both experiments, subjects were explicitly informed of the prior distribution at the beginning of each condition, and each condition included two preliminary training phases that familiarized them with the prior (the specifics for each task are detailed above).

To estimate the proportion of subjects best fit by the linear and the logarithmic encoding schemes, we conduct a ‘Bayesian model selection’ procedure (Rigoux et al., 2014) using the best-fitting model in each task. This procedure provides a Bayesian posterior over the proportion, as a beta distribution. In the estimation task, the posterior mean, indicating the expected proportion of logarithmic encoding, is 87.6% (as indicated in the main text), and the sum of the parameters of the beta distribution (indicating its concentration) is 38. In the discrimination task, the posterior mean is 45.9% (as indicated in the main text), and the sum of the parameters of the beta distribution is 65.

Here we consider an observer who has access to a series of i.i.d. signals, whose individual Fisher information over the support of the prior is $I_1(x)=A/x^2$. The constraint in Equation 5 determines A. We find

where $x_0$ and $x_1$ are the lower and upper bounds of the prior. We then substitute this expression of $I_1(x)$ in the optimization problem (Equation 6). Solving for $n$, we obtain the total Fisher information, as $I(x)=n{I}_1(x)$. With a uniform prior centered on $x_{\text{mid}}$, with $x_0=x_{\text{mid}}-h$ and $x_1=x_{\text{mid}}+h$, where $h$ is the half-width, we obtain:

Substituting $x_0$ and $x_1$ by their expressions as a function of $x_{\text{mid}}$ and $h$, we obtain the Taylor expansion:

that is at the first order the expressions in Equation 10, with $w=2h$, and $a=1$ in the estimation task and $a=2$ in the discrimination task.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Jessica Li and Maggie Lynn for their help as research assistants, Hassan Afrouzi for helpful comments, and the National Science Foundation for research support (grant SES DRMS 1949418).

Human subjects: The experiments complied with the relevant ethical regulations and were approved by Columbia University's Institutional Review Board (protocol numbers: IRB-AAAS8409 and IRB-AAAR9375).

1. Preprint posted: March 18, 2024
2. Sent for peer review: July 25, 2024
3. Reviewed Preprint version 1: September 20, 2024
4. Reviewed Preprint version 2: March 16, 2026
5. Reviewed Preprint version 3: March 17, 2026
6. Version of Record published: April 15, 2026

You can cite all versions using the DOI https://doi.org/10.7554/eLife.101277. This DOI represents all versions, and will always resolve to the latest one.

© 2024, Prat-Carrabin and Woodford

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