We consider a situation in which the objective magnitude of a stimulus with respect to some feature can be represented by a quantity ${\displaystyle v}$. When the stimulus is presented to an observer, it gives rise to an imprecise representation ${\displaystyle r}$ in the nervous system, on the basis of which the observer produces any required response. The internal representation ${\displaystyle r}$ can be stochastic, with given values being produced with conditional probabilities $p(r|v)$ that depend on the true magnitude. Here, we are more specifically concerned with discrimination experiments, in which two stimulus magnitudes $v_1$ and $v_2$ are presented, and the subject must choose which of the two is greater. We suppose that each magnitude $v_i$ has an internal representation $r_i$, drawn independently from a distribution $p(r_i|v_i)$ that depends only on the true magnitude of that individual stimulus. The observer’s choice must be based on a comparison of $r_1$ with $r_2$.

One way in which the cognitive resources recruited to make accurate discriminations may be limited is in the variety of distinct internal representations that are possible. When the complexity of feasible internal representations is limited, there will necessarily be errors in the identification of the greater stimulus magnitude in some cases, even assuming an optimal decoding rule for choosing the larger stimulus on the basis of $r_1$ and $r_2$. One can then consider alternative encoding rules for mapping objective stimulus magnitudes to feasible internal representations. The answer to this efficient coding problem generally depends on the prior distribution $f(v)$ from which the different stimulus magnitudes $v_i$ are drawn. The resources required for more precise internal representations of individual stimuli may be economized with respect to either or both of two distinct cognitive costs. The first goal of this work is to distinguish between these two types of efficiency concerns.

One question that we can ask is wheter the observed behavioral responses are consistent with the hypothesis that the conditional probabilities $p(r|v)$ are well-adapted to the particular frequency distribution of stimuli used in the experiment, suggesting an efficient allocation of the limited encoding neural resources. The assumption of full adaptation is typically adopted in efficient coding formulations of early sensory systems (Laughlin, 1981; Wei and Stocker, 2017), and also more recently in applications of efficient coding theories in value-based decisions (Louie and Glimcher, 2012; Polanía et al., 2019; Rustichini et al., 2017).

There is also a second cost in which it may be important to economize on cognitive resources. An efficient coding scheme in the sense described above economizes on the resources used to represent each individual new stimulus that is encountered; however, the encoding and decoding rules are assumed to be precisely optimized for the specific distribution $f(v)$ of stimuli that characterizes the experimental situation. In practice, it will be necessary for a decision maker to learn about this distribution in order to encode and decode individual stimuli in an efficient way, on the basis of experience with a given context. In this case, the relevant design problem should not be conceived as choosing conditional probabilities $p(r|v)$ once and for all, with knowledge of the prior distribution $f(v)$ from which ${\displaystyle v}$ will be drawn. Instead, it should be to choose a rule that specifies how the probabilities $p(r|v)$ should adapt to the distribution of stimuli that have been encountered in a given context. It then becomes possible to consider how well a given learning rule economizes on the degree of information about the distribution of magnitudes associated with one’s current context that is required for a given level of average performance across contexts. This issue is important not only to reduce the cognitive resources required to implement the rule in a given context (by not having to store or access so detailed a description of the prior distribution), but in order to allow faster adaptation to a new context when the statistics of the environment can change unpredictably (Młynarski and Hermundstad, 2019).

We now make the contrast between these two types of efficiency more concrete by considering a specific architecture for internal representations of sensory magnitudes. We suppose that the representation ${\displaystyle r_i}$ of a given stimulus will consist of the output of a finite collection of ${\displaystyle n}$ processing units, each of which has only two possible output states ('high' or 'low' readings), as in the case of a simple perceptron. The probability that each of the units will be in one output state or the other can depend on the stimulus ${\displaystyle v_i}$ that is presented. We further restrict the complexity of feasible encoding rules by supposing that the probability of a given unit being in the 'high’ state must be given by some function $\theta (v_i)$ that is the same for each of the individual units, rather than allowing the different units to coordinate in jointly representing the situation in some more complex way. We argue that the existence of multiple units operating in parallel effectively allows multiple repetitions of the same 'experiment’, but does not increase the complexity of the kind of test that can be performed. Note that we do not assume any unavoidable degree of stochasticity in the functioning of the individual units; it turns out that in our theory, it will be efficient for the units to be stochastic, but we do not assume that precise, deterministic functioning would be infeasible. Our resource limits are instead on the number of available units, the degree of differentiation of their output states, and the degree to which it is possible to differentiate the roles of distinct units.

Given such a mechanism, the internal representation $r_i$ of the magnitude of an individual stimulus $v_i$ will be given by the collection of output states of the ${\displaystyle n}$ processing units. A specification of the function $\theta (v)$ then implies conditional probabilities for each of the $2^n$ possible representations. Given our assumption of a symmetrical and parallel process, the number $k_i$ of units in the 'high' state will be a sufficient statistic, containing all of the information about the true magnitude $v_i$ that can be extracted from the internal representation. An optimal decoding rule will therefore be a function only of $k_i$, and we can equivalently treat $k_i$ (an integer between 0 and ${\displaystyle n}$) as the internal representation of the quantity $v_i$. The conditional probabilities of different internal representations are then

The efficient coding problem for a given environment, specified by a particular prior distribution $f(v),$ will be to choose the encoding rule $\theta (v)$ so as to allow an overall distribution of responses across trials that will be as accurate as possible (according to criteria that we will elaborate further below). We can further suppose that each of the individual processing units is a threshold unit, that produces a 'high’ reading if and only if the value $v_i-{\eta }_i$ exceeds some threshold $\tau ,$ where ${\eta }_i$ is a random term drawn independently on each trial from some distribution $f_{\eta }$ (Figure 1). The encoding function $\theta (v)$ can then be implemented by choosing an appropriate distribution $f_{\eta }$. This implementation requires that $\theta (v)$ be a non-decreasing function, as we shall assume.

Figure 1

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One measure of the cognitive resources required by such a system is the number ${\displaystyle n}$ of processing units that must produce an output each time an individual stimulus $v_i$ is evaluated. We can consider the optimal choice of $f_{\eta }$ in order to maximize, for instance, average accuracy of responses in a given environment $f(v)$, in the case of any bound ${\displaystyle n}$ on the number of units that can be used to represent each stimulus. But we can also consider the amount of information about the distribution $f(v)$ that must be used in order to decide how to encode a given stimulus $v_i$. If the system is to be able to adapt to changing environments, it must determine the value of ${\displaystyle \theta }$ (the probability of a 'high' reading) as a function of both the current $v_i$ and information about the distribution ${\displaystyle f}$, in a way that must now be understood to apply across different potential contexts. This raises the issue of how precisely the distribution ${\displaystyle f}$ associated with the current context is represented for purposes of such a calculation. A more precise representation of the prior (allowing greater sensitivity to fine differences in priors) will presumably entail a greater resource cost or very long adaptation periods.

We can quantify the precision with which the prior ${\displaystyle f}$ is represented by supposing that it is represented by a finite sample of ${\displaystyle m}$ independent draws ${\displaystyle {\tilde{v}}_1,\dots ,{\tilde{v}}_m}$ from the prior (or more precisely, from the set of previously experienced values, an empirical distribution that should after sufficient experience provide a good approximation to the true distribution). We further assume that an independent sample of ${\displaystyle m}$ previously experienced values is used by each of the processing units (Figure 1). Each of the ${\displaystyle n}$ individual processing units is then in the 'high' state with probability $\theta (v_i;{\tilde{v}}_1,\mathrm{\dots },{\tilde{v}}_m)$. The complete internal representation of the stimulus $v_i$ is then the collection of ${\displaystyle n}$ independent realizations of this binary-valued random variable. We may suppose that the resource cost of an internal representation of this kind is an increasing function of both ${\displaystyle n}$ and ${\displaystyle m}$.

This allows us to consider an efficient coding meta-problem in which for any given values $(n,m)$ the function $\theta (v_i;{\tilde{v}}_1,\mathrm{\dots },{\tilde{v}}_m)$ is chosen so as to maximize some measure of average perceptual accuracy, where the average is now taken not only over the entire distribution of possible $v_i$ occurring under a given prior $f(v),$ but over some range of different possible priors for which the adaptive coding scheme is to be optimized. We wish to consider how each of the two types of resource constraint (a finite bound on ${\displaystyle n}$ as opposed to a finite bound on ${\displaystyle m}$) affects the nature of the predicted imprecision in internal representations, under the assumption of a coding scheme that is efficient in this generalized sense, and then ask whether we can tell in practice how tight each of the resource constraints appears to be.

We first consider efficient coding in the case that there is no relevant constraint on the size of ${\displaystyle m}$, while ${\displaystyle n}$ instead is bounded. In this case, we can assume that each time an individual stimulus $v_i$ must be encoded, a large enough sample of prior values is used to allow accurate recognition of the distribution $f(v),$ and the problem reduces to a choice of a function $\theta (v)$ that is optimal for each possible prior $f(v).$

Maximizing mutual information

The nature of the resource-constrained problem to be optimized depends on the performance measure that we use to determine the usefulness of a given encoding scheme. A common assumption in the literature on efficient coding has been that the encoding scheme maximizes the mutual information between the true stimulus magnitude and its internal representation (Ganguli and Simoncelli, 2014; Polanía et al., 2019; Wei and Stocker, 2015). We start by characterizing the optimal $\theta (v)$ for a given prior distribution $f(v)$, according to this criterion. It can be shown that for large ${\displaystyle n}$, the mutual information between ${\displaystyle \theta }$ and ${\displaystyle k}$ (hence the mutual information between ${\displaystyle v}$ and ${\displaystyle k}$) is maximized if the prior distribution $\widehat{f}$ over ${\displaystyle \theta }$ is Jeffreys’ prior (Clarke and Barron, 1994)

(2) $\widehat{f}(\theta )=\frac{1}{\pi \sqrt{\theta (1-\theta )}},$

also known as the arcsine distribution. Hence, the mapping $\theta (v)$ induces a prior distribution $\widehat{f}$ over θ given by the arcsine distribution (Figure 2a, right panel). Based on this result, it can be shown that the optimal encoding rule $\theta (v)$ that guarantees maximization of mutual information between the random variable ${\displaystyle v}$ and the noisy encoded percept ${\displaystyle k}$ is given by (see Appendix 1)

(3) $\theta (v)={\left[\sin \left(\frac{\pi }{2}F(v)\right)\right]}^2,$

where $F(v)$ is the CDF of the prior distribution $f(v)$.

Figure 2

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In the previous section, we obtained analytical solutions that approximately characterize the optimal $\theta (v)$ in the limit as ${\displaystyle n}$ is made sufficiently large. Note however that we are always assuming that is finite, and that this constrains the accuracy of the decision maker’s judgments, while ${\displaystyle m}$ is instead unbounded and hence no constraint.

The nature of the optimal function $\theta (v_i;{\tilde{v}}_1,\mathrm{\dots },{\tilde{v}}_m)$ is different, however, when ${\displaystyle m}$ is small. We argue that this scenario is particularly relevant when full knowledge of the prior is not warranted given the costs vs benefits of learning, for instance, when the system expects contextual changes to occur often. In this case, as we will formally elaborate below, it ceases to be efficient for θ to vary only gradually as a function of $v_i$, rather than moving abruptly from values near zero to values near one (Appendix 6). In the large-${\displaystyle m}$ limiting case, the distributions of sample values $({\tilde{v}}_1,\mathrm{\dots },{\tilde{v}}_m)$ used by the different processing units will be nearly the same for each unit (approximating the current true distribution $f(v)$). Then if ${\displaystyle \theta }$ were to take only the values zero and one for different values of its arguments, the ${\displaystyle n}$ units would simply produce ${\displaystyle n}$ copies of the same output (either zero or one) for any given stimulus $v_i$ and distribution $f(v)$. Hence only a very coarse degree of differentiation among different stimulus magnitudes would be possible. Having ${\displaystyle \theta }$ vary more gradually over the range of values of $v_i$ in the support of $f(v)$ instead makes the representation more informative. But when ${\displaystyle m}$ is small (e.g., because of costs vs benefits of accurately representing the prior ${\displaystyle f}$), this kind of arbitrary randomization in the output of individual processing units is no longer essential. There will already be considerable variation in the outputs of the different units, even when the output of each unit is a deterministic function of $(v_i;{\tilde{v}}_1,\mathrm{\dots },{\tilde{v}}_m)$, owing to the variability in the sample of prior observations that is used to assess the nature of the current environment. As we will show below, this variability will already serve to allow the collective output of the several units to differentiate between many gradations in the magnitude of $v_i$, rather than only being able to classify it as 'small' or 'large' (because either all units are in the 'low’ or 'high’ states).

Our goal now is to compare back-to-back the resource-limited coding frameworks elaborated above in a fundamental cognitive function for human behavior: numerosity perception. We designed a set of experiments that allowed us to test whether human participants would adapt their numerosity encoding system to maximize fitness or accuracy rates via full prior adaptation as usually assumed in optimal models, or whether humans employ a 'less optimal' but more efficient strategy such as DbS, or the more established logarithmic encoding model.

In Experiment 1, healthy volunteers (n = 7) took part in a two-alternative forced choice numerosity task in which each participant completed ∼2400 trials across four consecutive days (Materials and methods). On each trial, they were simultaneously presented with two clouds of dots and asked which one contained more dots, and were given feedback on their reward and opportunity losses on each trial (Figure 3a). Participants were either rewarded for their accuracy (perceptual condition, where maximizing the amount of correct responses is the optimal strategy) or the number of dots they selected (value condition, where maximizing reward is the optimal strategy). Each condition was tested for two consecutive days with the starting condition randomized across participants. Crucially, we imposed a prior distribution $f(v)$ with a right-skewed quadratic shape (Figure 3b), whose parametrization allowed tractable analytical solutions of the encoding rules ${\theta }_A(v)$, ${\theta }_R(v)$ and ${\theta }_D(v)$, that correspond to the encoding rules for Accuracy maximization, Reward maximization, and DbS, respectively (Figure 3e and Materials and methods). Qualitative predictions of behavioral performance indicate that the accuracy-maximization model is the most accurate for trials with lower numerosities (the most frequent ones), whereas the reward-maximization model outperforms the others for trials with larger numerosities (trials where the difference in the number of dots in the clouds, and thus the potential reward, is the largest, Figure 2d and Figure 3f). In contrast, the DbS strategy presents markedly different performance predictions, in line with the discriminability predictions of our formal analyses (Figure 2c,d).

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In our modelling specification, the choice structure is identical for the three different sampling models, differing only in the encoding rule $\theta (v)$ (Materials and methods). Therefore, answering the question of which encoding rule is the most favored for each participant can be parsimoniously addressed using a latent-mixture model, where each participant uses ${\theta }_A(v)$, ${\theta }_R(v)$ or ${\theta }_D(v)$ to guide their decisions (Materials and methods). Before fitting this model to the empirical data, we confirmed the validity of our model selection approach through a validation procedure using synthetic choice data (Figure 3d, Figure 3—figure supplement 1, and Materials and methods).

After we confirmed that we can reliably differentiate between our competing encoding rules, the latent-mixture model was initially fitted to each condition (perceptual or value) using a hierarchical Bayesian approach (Materials and methods). Surprisingly, we found that participants did not follow the accuracy or reward optimization strategy in the respective experimental condition, but favored the DbS strategy (proportion that DbS was deemed best in the perceptual $p_{\mathrm{DbSfavored}}=0.86$ and value $p_{\mathrm{DbSfavored}}=0.93$ conditions, Figure 4). Importantly, this population-level result also holds at the individual level: DbS was strongly favored in 6 out of 7 participants in the perceptual condition, and seven out of seven in the value condition (Figure 4—figure supplement 1). These results are not likely to be affected by changes in performance over time, as performance was stable across the four consecutive days (Figure 4—figure supplement 2). Additionally, we investigated whether biases induced by choice history effects may have influenced our results (Abrahamyan et al., 2016; Keung et al., 2019; Talluri et al., 2018). Therefore, we incorporated both choice- and correctness-dependence history biases in our models and fitted the models once again (Materials and methods). We found similar results to the history-free models ($p_{\mathrm{DbSfavored}}=0.87$ in perceptual and $p_{\mathrm{DbSfavored}}=0.93$ in value conditions, Figure 4c). At the individual level, DbS was again strongly favored in 6 out of 7 participants in the perceptual condition, and 7 out of 7 in the value condition (Figure 4—figure supplement 1).

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In order to investigate further the robustness of this effect, we introduced a slight variation in the behavioral paradigm. In this new experiment (Experiment 2), participants were given points on each trial and had to reach a certain threshold in each run for it to be eligible for reward (Figure 3a and Materials and methods). This class of behavioral task is thought to be in some cases more ecologically valid than trial-independent choice paradigms (Kolling et al., 2014). In this new experiment, either a fixed amount of points for a correct trial was given (perceptual condition) or an amount equal to the number of dots in the chosen cloud if the response was correct (value condition). We recruited a new set of participants (n = 6), who were tested on these two conditions, each for two consecutive days with the starting condition randomized across participants (each participant completed $\sim 2,560$ trials). The quantitative results revealed once again that participants did not change their encoding strategy depending on the goals of the task, with DbS being strongly favored for both perceptual and value conditions ($p_{\mathrm{DbSfavored}}=0.999$ and $p_{\mathrm{DbSfavored}}=0.91$, respectively; Figure 4a), and these results were confirmed at the individual level where DbS was strongly favored in 6 out of 6 participants in both the perceptual and value conditions (Figure 4—figure supplement 1). Once again, we found that inclusion of choice history biases in this experiment did not significantly affect our results both at the population and individual levels. Population probability that DbS was deemed best in the perceptual ($p_{\mathrm{DbSfavored}}=0.999$) and value ($p_{\mathrm{DbSfavored}}=0.90$) conditions (Figure 4—figure supplement 1), and at the individual level DbS was strongly favored in 6 out of 6 participants in the perceptual condition and 5 of 6 in the value condition (Figure 4—figure supplement 1). Thus, Experiments 1 and 2 strongly suggest that our results are not driven by specific instructions or characteristics of the behavioral task.

As a further robustness check, for each participant we grouped the data in different ways across experiments (Experiments 1 and 2) and experimental conditions (perceptual or value) and investigated which sampling model was favored. We found that irrespective of how the data was grouped, DbS was the model that was clearly deemed best at the population (Figure 4) and individual level (Figure 4—figure supplement 3). Additionally, we investigated whether these quantitative results specifically depended on our choice of using a latent-mixture model. Therefore, we also fitted each model independently and compared the quality of the model fits based on out-of-sample cross-validation metrics (Materials and methods). Once again, we found that the DbS model was favored independently of experiment and conditions (Figure 4).

One possible reason why the two experimental conditions did not lead to differences could be that, after doing one condition for two days, the participants did not adapt as easily to the new incentive rule. However, note that as the participants did not know of the second condition before carrying it out, they could not adopt a compromise between the two behavioral objectives. Nevertheless, we fitted the latent-mixture model only to the first condition that was carried out by each participant. We found once again that DbS was the best model explaining the data, irrespective of condition and experimental paradigm (Figure 4—figure supplement 7). Therefore, the fact that DbS is favored in the results is not an artifact of carrying out two different conditions in the same participants.

We also investigated whether the DbS model makes more accurate predictions than the widely used logarithmic model of numerosity discrimination tasks (Dehaene, 2003). We found that DbS still made better out-of-sample predictions than the log-model (Figure 4b, Figure 5f,g). Moreover, these results continued to hold after taking into account possible choice history biases (Figure 4—figure supplement 4). In addition to these quantitative results, qualitatively we also found that behavior closely matched the predictions of the DbS model remarkably well (Figure 4c), based on virtually only one free parameter, namely, the number of samples (resources) ${\displaystyle n}$. Together, these results provide compelling evidence that DbS is the most likely resource-constrained sampling strategy used by participants in numerosity discrimination tasks.

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Recent studies have also investigated behavior in tasks where perceptual and preferential decisions have been investigated in paradigms with identical visual stimuli (Dutilh and Rieskamp, 2016; Polanía et al., 2014; Grueschow et al., 2015). In these tasks, investigators have reported differences in behavior, in particular in the reaction times of the responses, possibly reflecting differences in behavioral strategies between perceptual and value-based decisions. Therefore, we investigated whether this was the case also in our data. We found that reaction times did not differ between experimental conditions for any of the different performance assessments considered here (Figure 4—figure supplement 5). This further supports the idea that participants were in fact using the same sampling mechanism irrespective of behavioral goals.

Here it is important to emphasize that all sampling models and the logarithmic model of numerosity have the same degrees of freedom (performance is determined by ${\displaystyle n}$ in the sampling models and Weber’s fraction ${\displaystyle \sigma }$ in the log model, Materials and methods). Therefore, qualitative and quantitative differences favoring the DbS model cannot be explained by differences in model complexity. It could also be argued that normal approximation of the binomial distributions in the sampling decision models only holds for large enough ${\displaystyle n}$. However, we find evidence that the large-${\displaystyle n}$ optimal solutions are also nearly optimal for low ${\displaystyle n}$ values (Appendix 7). Estimates of ${\displaystyle n}$ in our data are in general $n\approx 21$ (Table 1) and we find that the large-${\displaystyle n}$ rule is nearly optimal already for $n=15$ (Appendix 7). Therefore the asymptotic approximations should not greatly affect the conclusions of our work.

Up to now, fits and comparison across models have been done under the assumption that the participants learned the prior distribution $f(v)$ imposed in our task. If participants are employing DbS, it is important to understand the dynamical nature of adaptation in our task. Note that the shape of the prior distribution is determined by the parameter ${\displaystyle \alpha }$ (Figure 5b, Equation 10 in Materials and methods). First, we made sure based on model recovery analyses that the DbS model could jointly and accurately recover both the shape parameter ${\displaystyle \alpha }$ and the resource parameter ${\displaystyle n}$ based on synthetic data (Figure 3—figure supplement 2). Then we fitted this model to the empirical data and found that the recovered value of the shape parameter ${\displaystyle \alpha }$ closely followed the value of the empirical prior with a slight underestimation (Figure 5a). Next, we investigated the dynamics of prior adaptation. To this end, we ran a new experiment (Experiment 3, n = 7 new participants) in which we set the shape parameter of the prior to a lower value compared to Experiments 1–2 (Figure 5b, Materials and methods). We investigated the change of ${\displaystyle \alpha }$ over time by allowing this parameter to change with trial experience (Equation 18, Materials and methods) and compared the evolution of ${\displaystyle \alpha }$ for Experiments 1 and 2 (empirical $\alpha =2$) with Experiment 3 (empirical $\alpha =1$, Figure 5b). If participants show prior adaptation in our numerosity discrimination task, we hypothesized that the asymptotic value of ${\displaystyle \alpha }$ should be higher for Experiments 1–2 than for Experiment 3. First, we found that for Experiments 1–2, the value of ${\displaystyle \alpha }$ quickly reached an asymptotic value close to the target value (Figure 5c). On the other hand, for Experiment 3 the value of ${\displaystyle \alpha }$ continued to decrease during the experimental session, but slowly approaching its target value. This seemingly slower adaptation to the shape of the prior in Experiment 3 might be explained by the following observation. The prior parametrized with $\alpha =1$ in Experiment 3 is further away from an agent hypothesized to have a natural numerosity discrimination based on a log scale ($\alpha =2.58$, Materials and methods), which is closer in value to the shape of the prior in Experiments 1 and 2 ($\alpha =2$). Irrespective of these considerations, the key result to confirm our adaptation hypothesis is that the asymptotic value of ${\displaystyle \alpha }$ is lower for Experiment 3 compared to Experiments 1 and 2 ($P_{\mathrm{MCMC}}=0.006$).

In order to make sure that this result was not an artifact of the parametric form of adaptation assumed here (Equation 18, Materials and methods), we fitted the DbS model to trials at the beginning and end of each experimental session allowing ${\displaystyle \alpha }$ to be a free but fixed parameter in each set of trials. The results of these new analyses are virtually identical to the results obtained with the parametric form, in which ${\displaystyle \alpha }$ is smaller at the end of Experiment 3 sessions relative to beginning of Experiments 1 and 2 ($P_{\mathrm{MCMC}}=0.0003$), beginning of Experiments 3 ($P_{\mathrm{MCMC}}=0.013$) and end of Experiments 1 and 2 ($P_{\mathrm{MCMC}}=0.006$, Figure 5d). In this model, we did not allow ${\displaystyle n}$ to freely change for each condition, and therefore a concern might be that the results might be an artifact of changes in ${\displaystyle n}$, which could for example change with the engagement of the participants across the session. Given that we already demonstrated that both parameters ${\displaystyle n}$ and ${\displaystyle \alpha }$ are identifiable, we fitted the same model as in Figure 5d, however this time we allowed ${\displaystyle n}$ to be free parameter alongside ${\displaystyle \alpha }$. We found that the results obtained in Figure 5d remained virtually unchanged (Figure 5—figure supplement 3), in addition to the result that the resource parameter ${\displaystyle n}$ remained virtually identical across the session (Figure 5—figure supplement 3).

We further investigated evidence for adaptation using an alternative quantitative approach. First, we performed out-of-sample model comparisons based on the following models: (i) the adaptive-${\displaystyle \alpha }$ model, (ii) free-${\displaystyle \alpha }$ model with ${\displaystyle \alpha }$ free but non-adapting over time, and (iii) fixed-${\displaystyle \alpha }$ model with $\alpha =2$. The results of the out-of-sample predictions revealed that the best model was the free-${\displaystyle \alpha }$ model, followed closely by the adaptive-${\displaystyle \alpha }$ model ($\mathrm{\Delta }\mathrm{LOO}=1.8$) and then by fixed-${\displaystyle \alpha }$ model ($\mathrm{\Delta }\mathrm{LOO}=32.6$). However, we did not interpret the apparent small difference between the adaptive-${\displaystyle \alpha }$ and the free-${\displaystyle \alpha }$ models as evidence for lack of adaptation, given that the more complex adaptive-${\displaystyle \alpha }$ model will be strongly penalized after adaptation is stable. That is, if adaptation is occurring, then the adaptive-${\displaystyle \alpha }$ only provides a better fit for the trials corresponding to the adaptation period. After adaptation, the adaptive-${\displaystyle \alpha }$ should provide a similar fit than the free-${\displaystyle \alpha }$ model, however with a larger complexity that will be penalized by model comparison metrics. Therefore, to investigate the presence of adaptation, we took a closer quantitative look at the evolution of the fits across trial experience. We computed the average trial-wise predicted Log-Likelihood (by sampling from the hierarchical Bayesian model) and compared the differences of this metric between the competing models and the adaptive model. We hypothesized that if adaptation is taking place, the adaptive-${\displaystyle \alpha }$ model would have an advantage relative to the free-${\displaystyle \alpha }$ model at the beginning of the session, with these differences vanishing toward the end. On the other hand, the fixed-${\displaystyle \alpha }$ should roughly match the adaptive-${\displaystyle \alpha }$ model at the beginning and then become worse over time, but these differences should stabilize after the end of the adaptation period. The results of these analyses support our hypotheses (Figure 5—figure supplement 2), thus providing further evidence of adaptation, highlighting the fact that the DbS model can parsimoniously capture adaptation to contextual changes in a continuous and dynamical manner. Furthermore, we found that the DbS model again provides more accurate qualitative and quantitative out-of-sample predictions than the log model (Figure 5e,f).

The study tested young healthy volunteers with normal or corrected-to-normal vision (total n = 20, age 19–36 years, nine females: n = 7 in Experiment 1, two females; n = 6 new participants in Experiment 2, three females; n = 7 new participants in Experiment 3, four females). Participants were randomly assigned to each experiment and no participant was excluded from the analyses. Participants were instructed about all aspects of the experiment and gave written informed consent. None of the participants suffered from any neurological or psychological disorder or took medication that interfered with participation in our study. Participants received monetary compensation for their participation in the experiment partially related to behavioral performance (see below). The experiments conformed to the Declaration of Helsinki and the experimental protocol was approved by the Ethics Committee of the Canton of Zurich (BASEC: 2018–00659).

Participants (n = 7) carried out a numerosity discrimination task for four consecutive days for approximately one hour per day. Each daily session consisted of a training run followed by 8 runs of 75 trials each. Thus, each participant completed ∼2400 trials across the four days of experiment.

After a fixation period (1–1.5 s jittered), two clouds of dots (left and right) were presented on the screen for 200 ms. Participants were asked to indicate the side of the screen where they perceived more dots. Their response was kept on the screen for 1 s followed by feedback consisting of the symbolic number of dots in each cloud as well as the monetary gains and opportunity losses of the trial depending on the experimental condition. In the value condition, participants were explicitly informed that each dot in a cloud of dots corresponded to 1 Swiss Franc (CHF). Participants were informed that they would receive the amount in CHF corresponding to the total number of dots on the chosen side. At the end of the experiment a random trial was selected and they received the corresponding amount. In the accuracy condition, participants were explicitly informed that they could receive a fixed reward (15 Swiss Francs (CHF)) for each correct trial. This fixed amount was selected such that it approximately matched the expected reward received in the value condition (as tested in pilot experiments). At the end of the experiment, a random trial was selected and they would receive this fixed amount if they chose the cloud with more dots (i.e., the correct side). Each condition lasted for two consecutive days with the starting condition randomized across participants. Only after completing all four experiment days, participants were compensated for their time with 20 CHF per hour, in addition to the money obtained based on their decisions on each experimental day.

Participants (n = 6) carried out a numerosity discrimination task in which each of four daily sessions consisted of 16 runs of 40 trials each, thus each participant completed ∼2560 trials. A key difference with respect to Experiment 1 is that participants had to accumulate points based on their decisions and had to reach a predetermined threshold on each run. The rules of point accumulation depended on the experimental condition. In the perceptual condition, a fixed amount of points was awarded if the participants chose the cloud with more dots. In this condition, participants were instructed to accumulate a number of points and reach a threshold given a limited number of trials. Based on the results obtained in Experiment 1, the threshold corresponded to 85% of correct trials in a given run, however the participants were unaware of this. If the participants reached this threshold, they were eligible for a fixed reward (20 CHF) as described in Experiment 1. In the value condition, the number of points received was equal to the number of dots in the cloud, however, contrary to Experiment 1, points were only awarded if the participant chose the cloud with the most dots. Participants had to reach a threshold that was matched in the expected collection of points of the perceptual condition. As in Experiment 1, each condition lasted for two consecutive days with the starting condition randomized across participants. Only after completing all the four days of the experiment, participants were compensated for their time with 20 CHF per hour, in addition to the money obtained based on their decisions on each experimental day.

The design of Experiment 3 was similar to the value condition of Experiment 2 (n = 7 participants) and was carried out over three consecutive days. The key difference between Experiment 3 and Experiments 1–2 was the shape of the prior distribution $f(v)$ that was used to draw the number of dots for each cloud in each trial (see below).

For all experiments, we used the following parametric form of the prior distribution

initially defined in the interval [0,1] for mathematical tractability in the analytical solution of the encoding rules $\theta (v)$ (see below), with $\alpha >0$ determining the shape of the distribution, and ${\displaystyle c}$ is a normalizing constant. For Experiments 1 and 2 the shape parameter was set to $\alpha =2$, and for Experiment 3 was set to $\alpha =1$. i.i.d. samples drawn from this distribution were then multiplied by 50, added an offset of 5, and finally were rounded to the closest integer (i.e., the numerosity values in our experiment ranged from $v_{\min }=5$ to $v_{\max }=55$). The pairs of dots on each trial were determined by sampling from a uniform density window in the CDF space (Equation 10 is its corresponding PDF). The pairs of dots in each trial were selected with the conditions that, first, their distance in the CDF space was less than a constant (0.25, 0.28 and 0.23 for Experiments 1, 2 and 3 respectively), and second, the number of dots in both clouds was different. Figure 3c illustrates the probability that a pair of choice alternatives was selected for a given trial in Experiments 1 and 2.

Given that adaptation dynamics in sensory systems often require long-term experience with novel prior distributions, we opted for maximizing the number of trials for a relatively small number of participants per experiment, as it is commonly done for this type of psychophysical experiments (Brunton et al., 2013; Stocker and Simoncelli, 2006; Zylberberg et al., 2018). Note that based on the power analyses described below, we collected in total ∼45,000 trials across the three Experiments, which is above the average number of trials typically collected in human studies.

In order to maximize statistical power in the differentiation of the competing encoding rules, we generated 10,000 sets of experimental trials for each encoding rule and selected the sets of trials with the highest discrimination power (i.e., largest differences in Log-Likelihood) between the encoding models. In these power analyses, we also investigated what was the minimum number of trials that would allow accurate generative model selection at the individual level. We found that ∼1000 trials per participant in each experimental condition would be sufficient to predict accurately (P>0.95) the true generative model. Based on these analyses, we decided to collect at least 1200 trials per participant and condition (perceptual and value) in each of the three experiments. Model recovery analyses presented in Figure 3d illustrate the result of our power analyses (see also Figure 3—figure supplement 1).

Eyetracking (EyeLink 1000 Plus) was used to check the participants' fixation during stimulus presentation. When participants blinked or moved their gaze (more than 2° of visual angle) away from the fixation cross during the stimulus presentation, the trial was canceled (only 212 out of 45,600 trials were canceled, that is, <0.5% of the trials). Participants were informed when a trial was canceled and were encouraged not to do so as they would not receive any reward for this trial. A chinrest was used to keep the distance between the participants and the screen constant (55 cm). The task was run using Psychtoolbox Version 3.0.14 on Matlab 2018a. The diameter of the dots varied between 0.42° and 1.45° of visual angle. The center of each cloud was positioned 12.6° of visual angle horizontally from the fixation cross and had a maximum diameter of 19.6° of visual angle. Following previous numerosity experiments (van den Berg et al., 2017; Izard and Dehaene, 2008), either the average dot size or the total area covered by the dots was maintained constant in both clouds for each trial. The color of each dot (white or black) was randomly selected for each dot. Stimuli sets were different for each participant but identical between the two conditions.

The parametrization of the prior $f(v)$ (Equation 10) allows tractable analytical solutions of the encoding rules ${\theta }_A(v)$, ${\theta }_R(v)$ and ${\theta }_D(v)$, that correspond to Accuracy maximization, Reward maximization, and DbS, respectively:

Graphical representation of the respective encoding rules is shown in Figure 3e for Experiments 1 and 2. Given an encoding rule $\theta (v)$, we now define the decision rule. The goal of the decision maker in our task is always to decide which of two input values $v_1$ and $v_2$ is larger. Therefore, the agent choses $v_1$ if and only if the internal readings $k_1>k_2$. Following the definitions of expected value and variance of binomial variables, and approximating for large ${\displaystyle n}$ (see Appendix 2), the probability of choosing $v_1$ is given by

where $\mathrm{\Phi }()$ is the standard CDF, and ${\theta }_1$ and ${\theta }_2$ are the encoding rules for the input values $v_1$ and $v_2$, respectively. Thus, the choice structure is the same for all models, only differing in their encoding rule. The three models generate different qualitative performance predictions for a given number of samples ${\displaystyle n}$ (Figure 3f).

Crucially, this probability decision rule (Equation 14) can be parsimoniously extended to include potential side biases independent of the encoding process as follows

where ${\beta }_0$ is the bias term. This is the base model used in our work. We were also interested in studying whether choice history effects (Abrahamyan et al., 2016; Talluri et al., 2018) may have influence in our task, thus possibly affecting the conclusions that can be drawn from the base model. Therefore, we extended this model to incorporate the effect of decision learning and choices from the previous trial

where $a_{t-1}$ is the choice made on the previous trial (+1 for left choice and −1 for right choice) and $r_{t-1}$ is the 'outcome learning' on the previous trial (+1 for correct choice and −1 for incorrect choice). ${\beta }^{\mathrm{L}}$ and ${\beta }^{\mathrm{Ch}}$ capture the effect of decision learning and choice in the previous trial, respectively.

Given that the choice structure is the same for all three sampling models considered here, we can naturally address the question of what decision rule the participants favor via a latent-mixture model. We implemented this model based on a hierarchical Bayesian modelling (HBM) approach. The base-rate probabilities for the three different encoding rules at the population level are represented by the vector ${\displaystyle \mathit{\pi }}$, so that ${\displaystyle {\mathit{\pi }}_{\mathit{m}}}$ is the probability of selecting encoding rule model ${\displaystyle m}$. We initialize the model with an uninformative prior given by

This base-rate is updated based on the empirical data, where we allow each participant ${\displaystyle s}$ to draw from each model categorically based on the updated base-rate

where the encoding rule ${\displaystyle \theta }$ for model ${\displaystyle m}$ is given by

The selected rule was then fed into Equations 15 and 16 to determine the probability of selecting a cloud of dots. The number of samples ${\displaystyle n}$ was also estimated within the same HBM with population mean ${\displaystyle \mu }$${\displaystyle \mu }$ and standard deviation ${\displaystyle \sigma }$ initialized based on uninformative priors with plausible ranges

allowing each participant ${\displaystyle s}$ to draw from this population prior assuming that ${\displaystyle n}$ is normally distributed at the population level

Similarly, the latent variables ${\displaystyle \beta }$ in equations Equations 15 and 16 were estimated by setting population mean ${\mu }_{\beta }$ and standard deviation ${\sigma }_{\beta }$ initialized based on uninformative priors

allowing each participant ${\displaystyle s}$ to draw from this population prior assuming that ${\displaystyle \beta }$ is normally distributed at the population level

In all the results reported in Figure 3 and Figure 4, the value of the shape parameter of the prior was set to its true value $\alpha =2$. The estimation of ${\displaystyle \alpha }$ in Figure 5a was investigated with a similar hierarchical approach, allowing each participant to sample from the normal population distribution with uninformative priors over the population mean and standard deviation

The choice rule of the standard logarithmic model of numerosity discrimination is given by

where ${\displaystyle \sigma }$ is the internal noise in the logarithmic space. This model was extended to incorporate bias and choice history effects in the same way as implemented in the sampling models. Here, we emphasize that all sampling and log models have the same degrees of freedom, where performance is mainly determined by ${\displaystyle n}$ in the sampling models and Weber’s fraction ${\displaystyle \sigma }$ in the log model, and biases are determined by parameters ${\displaystyle \beta }$. For all above-mentioned models, the trial-by-trial likelihood of the observed choice (i.e., the data) given probability of a decision was based on a Bernoulli process

where $y_{t,s}\in \{0,1\}$ is the decision of each participant ${\displaystyle s}$ in each trial ${\displaystyle t}$. In order to allow for prior adaptation, the model fits presented in Figure 3 and Figure 4 were fit starting after a fourth of the daily trials (corresponding to 150 trials for Experiment 1 and 160 trials for Experiment 2) to allow for prior adaptation and fixing the shape parameter to its true generative value $\alpha =2$.

The dynamics of adaptation (Figure 5) were studied by allowing the shape parameter α to evolve through trial experience using all trials collected on each experiment day. This was studied using the following function

where ${\displaystyle \delta }$ represents a possible target adaptation value of ${\displaystyle \alpha }$, ${\displaystyle t}$ is the trial number, and ${\displaystyle \eta }$, ${\displaystyle \tau }$ determine the shape of the adaptation. Therefore, the encoding rule of the DbS model also changed trial-to-trial

Adaptation was tested based on the hypothesis that participants initially use a logarithmic discrimination rule (Equation 17) (this strategy also allowed improving identification of the adaptation dynamics). Therefore, Equation 18 was parametrized such that the initial value of the shape parameter (${\alpha }_{t=0}$) guaranteed that discriminability between the DbS and the logarithmic rule was as close as possible. This was achieved by finding the value of α in the DbS encoding rule (${\theta }_D$) that minimizes the following expression

where $v_{1,t}$ and $v_{2,t}$ are the numerosity inputs for each trial ${\displaystyle t}$. This expression was minimized based on all trials generated in Experiments 1–3 (note that minimizing this expression does not require knowledge of the sensitivity levels ${\displaystyle \sigma }$ and ${\displaystyle n}$ for the log and DbS models, respectively). We found that the shape parameter value that minimizes Equation 20 is $\alpha =2.58$. Based on our prior $f(v)$ parametrization (Equation 10), this suggests that the initial prior is more skewed than the priors used in Experiments 1–3 (Figure 5b). This is an expected result given that log-normal priors, typically assumed in numerosity tasks, are also highly skewed. We fitted the ${\displaystyle \delta }$ parameter independently for Experiments 1–2 and Experiment 3 but kept the ${\displaystyle \tau }$ parameter shared across all experiments. If adaptation is taking place, we hypothesized that the asymptotic value ${\displaystyle \delta }$ of the shape parameter ${\displaystyle \alpha }$ should be larger for Experiments 1–2 compared to Experiment 3.

Posterior inference of the parameters in all the hierarchical models described above was performed via the Gibbs sampler using the Markov Chain Monte Carlo (MCMC) technique implemented in JAGS. For each model, a total of 50,000 samples were drawn from an initial burn-in step and subsequently a total of new 50,000 samples were drawn for each of three chains (samples for each chain were generated based on a different random number generator engine, and each with a different seed). We applied a thinning of 50 to this final sample, thus resulting in a final set of 1000 samples for each chain (for a total of 3000 pooling all three chains). We conducted Gelman–Rubin tests for each parameter to confirm convergence of the chains. All latent variables in our Bayesian models had $\widehat{R}<1.05$, which suggests that all three chains converged to a target posterior distribution. We checked via visual inspection that the posterior population level distributions of the final MCMC chains converged to our assumed parametrizations. When evaluating different models, we are interested in the model's predictive accuracy for unobserved data, thus it is important to choose a metric for model comparison that considers this predictive aspect. Therefore, in order to perform model comparison, we used a method for approximating leave-one-out cross-validation (LOO) that uses samples from the full posterior (Vehtari et al., 2017). These analyses were repeated using an alternative Bayesian metric: the WAIC (Vehtari et al., 2017).