Manipulation check.

We first checked whether the draw-by-draw probability estimates for the hidden box reported by participants indicated that they generally engaged in the task as we expected. Indeed, averaging across all sequences and participants, probability estimates showed a gradual increase towards higher probabilities for the true hidden box as the number of observed beads increased, and the rate of this increase was higher for bead-ratio conditions denoting stronger evidence ([90:10]>[60:40]>[51:49]; interaction of bead draw [0–8] by bead-ratio condition: t_160.66 = 26.15, p = 8.82x10^-60, linear mixed-effects model; Fig 3A and S4 Table), suggesting that participants’ beliefs tracked the cumulative evidence strength of observed samples over a trial. This effect was also obvious in most individual participants and in an analysis restricted to a subset of 16 identical sequences (see Methods) across 60:40 and 90:10 bead-ratio conditions ([90:10] > [60:40]; interaction of bead draw [0–8] by bead-ratio condition: t_182.96 = 11.81, p = 2.79x10^-24, linear mixed-effects model; Fig 3A inset and S5 Table). The first estimates before seeing any bead were generally unbiased S3 Fig and no systematic between-trial effects were apparent.

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Fig 3. Study 1 participants show behavioral signatures of sequential base-rate neglect which scale with model-derived prior underweighting.

(a) Group mean of average probability estimates over bead draws for each bead-ratio condition. Participants updated beliefs progressively toward the correct hidden box with steeper slopes for stronger evidence. The inset shows the same data limited to matched (identical) sequences for the 60:40 and 90:10 conditions. Solid lines and shaded regions reflect the mean and standard error of the mean (SEM) of the weighted Bayesian model fits across participants. (b) Group mean of final estimate difference as a function of evidence asymmetry. Each data point shows the difference in the probability estimate after 8 beads for a back-loaded and a front-loaded sequence comprising a mirror-opposite pair, with positive values indicating higher estimates for back-loaded sequences consistent with recency bias. Solid lines and shaded regions reflect the mean and SEM of the weighted Bayesian model fits. Consistent with model predictions (Fig 2B), the data shows a recency bias scaling with evidence asymmetry. (c) Group median of individual medians for the magnitude of logit-belief updates as a function of the logit prior with respect to the color of the current bead sample, divided by bead-ratio condition. The x-axis is discretized into bins equivalent to 0.1 increments of the prior belief in probability space (with a lower limit of 0.01 and an upper limit of 0.99; data only binned for visualization). The y-axis represents the magnitude of the logit-belief updates (the difference in the log-odds of the prior and posterior beliefs). Solid lines and shaded regions reflect medians and 95% bootstrapped confidence intervals of the weighted Bayesian model fits. Although not displayed for visual clarity, the confidence intervals for the raw data overlap substantially with the model fits. For visualization only, we excluded extreme outlier or noisy data points (logit belief updates > 2, individual median values based on less than 3 data points for a given bin, group median values based on less than 25% of individuals) for a total exclusion of 6.96% of the data. Consistent with model predictions (Fig 2C), the data shows prior-dependent belief-updating with less updating for prior-consistent evidence (right of the vertical dashed line; i.e. an overall negative slope). Note that at the group level this effect appears to be non-monotonic (with slightly positive slope towards the rightmost end) due to aggregation of data across individuals with different ω₁ values, since individuals with ω₁ > 1 are predicted to have and exhibit more updating to prior-consistent evidence (i.e., positive slopes; S2 Fig). (d) Formal model comparison for data from study 1. We compared 10 different models as in our previous work [31]. Each model is defined by its free parameters, which are reflected on the x-axis. See S28 Table for details. The winning model was defined as the model with the highest protected exceedance probability, which was the same as in our previous work [31] and in study 2 (S6 Fig). (e) The evidence asymmetry slope (equivalent to a single line fitted across all conditions in panel b) is plotted against the prior-weight ω₁, showing a negative correlation. This correlation closely follows model predictions indicated by the black line (as in Fig 2D but with shaded regions including variability in likelihood-weight ω₂ parameters between the 25^th and 75^th percentile range of observed values in our previous work [31]). Marginal violin plots show group medians and interquartile ranges. (f) The mean final estimate difference is shown against ω₁, again showing a correlation that follows the model prediction (black line as in Fig 2E). Marginal violin plots show group medians and interquartile ranges. (e, f) Asterisks indicate a significant sign-rank tests of group medians against the corresponding reference values indicated by the dashed lines. Note that results in (e) and (f) were robust to the exclusion of outliers with an ω₁ more than 3 scaled median absolute deviations [52] from the median (ω₁<0.75; 11 outliers): after their removal, the correlation between ω₁ and the evidence asymmetry slope was still significant (ρ = -0.58, p < 10⁻³⁰⁷), as was the correlation between ω₁ and the mean final estimate difference (ρ = -0.53, p = 2.32 x 10⁻¹²). Posterior predictive checks further recapitulate the range of values in the data (S10–S12 Figs).

https://doi.org/10.1371/journal.pcbi.1010796.g003

Behavioral signatures of sequential base-rate neglect.

As mentioned above, the weighted Bayesian model predicts that base-rate neglecters (ω₁ < 1) should have a recency bias. In the sequential context relevant here, the recency bias should manifest at the end of a sequence as higher final probability estimates for the true hidden box when more majority beads (beads whose color is consistent with the true identity of the hidden box) are presented towards the end versus the beginning of the sequence (“back-loaded” versus “front-loaded” sequences, respectively). This would directly show that more recent samples, closer to the end of the sequence, have a stronger influence on the final estimate relative to older samples closer to the beginning (Fig 2A). Furthermore, model simulations showed that this effect should be more apparent when comparing pairs of sequences with more extreme front-loading and back-loading (Fig 2B), which we quantified based on a linear weighted sum of majority beads in the sequence based on their order (1^st to 8^th position) and which we refer to as ‘evidence asymmetry’ (with respect to the middle of the sequence). In contrast to a base-rate neglecter (prior weight ω₁ < 1), the Bayesian ideal observer exhibits path- or evidence-order-independence in its final beliefs, as do observers with different likelihood weighting (ω₂ ≠ 1; S2 Fig).

Our task design included a systematic sequence-level manipulation of evidence order and asymmetry (Fig 1B) to allow for a direct demonstration of recency bias. Per the above explanation, a simple test for this bias consisted of comparing the final probability estimate between mirror-opposite sequence pairs that had the same number of majority beads and bead ratio but in which majority beads were either front-loaded or back-loaded (Fig 2A). Critically, the data showed evidence-order-dependence in the form of a recency bias consistent with sequential base-rate neglect: pair-wise differences in final probability estimates were higher for back-loaded versus front-loaded sequences (mean final estimate difference > 0, p = 1.66x10^-8; sign-rank test) and this positive difference increased with evidence asymmetry (evidence asymmetry slope > 0, p = 2.22x10^-11; sign-rank test), with steeper slopes for stronger evidence ([90:10]>[60:40]>[51:49] bead-ratio conditions; bead ratio x evidence asymmetry interaction: t_151.46 = 3.107, p = 0.002, linear mixed-effects model; Fig 3B and S6 Table). All three findings conformed with the predictions of the sequential base-rate-neglect model.

A further prediction of the weighted Bayesian model is that sequential base-rate neglect induces a form of prior-dependent belief updating whereby, as the prior increases in favor of one option, the magnitude of logit belief updates to prior-consistent evidence decreases, and it increases to prior-inconsistent evidence (Fig 2C). This impedes reaching full certainty in beliefs over the long run, resulting in more “moderate” beliefs [1,2,17]. In contrast, the ideal observer would predict belief updates of constant magnitude in logit space. In line with sequential base-rate neglect and our model predictions, we observed that mean logit belief updates in response to prior-consistent evidence tended to decrease as prior certainty increased (logit-prior main effect: t_150.30 = -2.643, p = 0.009; Fig 3C), an effect which was independent of the bead-ratio condition (logit-prior x bead-ratio interaction: t_143.13 = 0.903, p = 0.368; linear mixed-effects model; S7 Table).

Thus, these model-agnostic results show evidence-order dependence and prior-dependent updating that are generally consistent with the predictions of sequential base-rate neglect under the weighted Bayesian model [1,2,17,31] and are satisfactorily captured by this model based on posterior predictive checks (shaded regions in Fig 3A–3C).

Relationship between model-agnostic and model-based measures of sequential base-rate neglect.

We carried out a group-level model comparison of variants of Bayesian-inference models, including the (unweighted) Bayesian ideal-observer model, as in previous work [31] (see Methods). As in this previous work, the winning model (Fig 3D; S8 Table) was the weighted Bayesian model with a prior-weight parameter (ω₁) and one likelihood-weight parameter per condition (, where (l) is one of the three bead-ratio conditions). Examining the fitted prior-weight ω₁ parameter values across participants revealed substantial interindividual variability and a general tendency for underweighting of prior beliefs (ω₁<1: p = 2.27x10^-4, sign-rank test), consistent with sequential base-rate neglect. Critically, and consistent with the model predictions (Fig 2D and 2E), participants exhibiting lower ω₁ values tended to exhibit stronger recency biases and stronger modulation with increasing evidence asymmetry in their final probability estimates (mean final estimate difference: ρ = −0.60, p<2.22 x 10⁻³⁰⁸; evidence asymmetry slope: ρ = −0.64, p<2.22 x 10⁻³⁰⁸; Spearman correlation; Fig 3E and 3F). The evidence asymmetry slope and the mean final estimate difference also correlated strongly with each other (ρ = −0.63, p<2.22 x 10⁻³⁰⁸). Note that the model-predicted relationship between the prior weight ω₁ and these model-agnostic measures of recency bias (Fig 2D and 2E) is non-monotonic for very low values of ω₁ but monotonic for the range of ω₁ values roughly over 0.75, where the majority of our data are (92.72%); results held when analyses were restricted to this monotonic range (see Fig 3 caption). Furthermore, prior-dependent updating—the slope of the logit belief update in the direction of the evidence as a function of the logit prior—across all bead-ratio conditions positively correlated with ω₁ (ρ = 0.71, p<2.22 x 10⁻³⁰⁸), and negatively correlated with the evidence-asymmetry effect (ρ = −0.50, p = 1.32 x 10⁻¹⁰) and the mean final estimate difference (ρ = −0.49, p = 2.57 x 10⁻¹⁰). These model-predicted relationships all held when controlling for the three parameters, and the model root-mean-squared error (RMSE; S9 Table) and were robust to exclusion of potential outliers (see Fig 3 caption).

This indicates consistency across model-based and model-agnostic analyses and highlights the specificity of the relationship between ω₁ and the predicted behavioral signatures of sequential base-rate neglect. Furthermore, measures of general cognition [47] and psychopathology [48] did not show a specific relationship with ω₁, suggesting that variability in ω₁ is unlikely to reflect domain-general factors (despite sufficient variability in both; S4 Fig and S10–13 Tables).

Relationship between laboratory indices of sequential base-rate neglect and odd real-world beliefs.

Individuals with more extreme sequential base-rate neglect may tend to hold peculiar beliefs due to excessive susceptibility to new evidence (i.e., recency bias) combined with an inability to resolve belief uncertainty [2] (per prior-dependent belief updating). To examine the relevance of interindividual variability in the task-based measures of sequential base-rate neglect to real-world beliefs, we collected a self-report questionnaire that measures proclivity to various odd or unusual beliefs (Peters Delusions Inventory [PDI] [49]; Methods). We did not observe significant relationships between the relevant measures of sequential base-rate neglect and PDI scores (S9 Table). However, very few participants had high PDI scores based on previously published cutoffs [50,51] (only 1–15 participants or ~1–10% of the sample), thus limiting our power to detect relationships with PDI. To address this, we conducted a second study that used pre-screening to ensure an adequate range of PDI scores.

Pre-screening, exclusions, and retained sample.

To ensure a wide range of PDI scores and sufficient high PDI participants with odd beliefs, Study 2 used a pre-screening procedure following prior work [53–56] (Methods). The study consisted of two parts: (i) a pre-screening based on the PDI (and, secondarily, on the Paranoia Checklist; Methods), and (ii) a separate experimental session involving administration of a second PDI and the task discussed above (separated on average by 3.5 days). Critically, the pre-screening used unbiased PDI-score cutoffs derived from previously published norms [49] (under 34.9 for low PDI and over 82.9 for high PDI; Methods). After exclusions (Methods), 116 participants were retained of whom 91 comprised the main sample: 34 in the high PDI group and 57 in the low PDI group (S1 Table; Fig 4A and 4B inset). Attesting to the effectiveness of (and need for) the pre-screening, note that only 2 participants from study 1 would have been classified as high PDI based on study 2’s pre-screening cutoffs.

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Fig 4. Replication in study 2 of results from study 1.

(a, b) Mean final estimate difference as a function of evidence asymmetry for the low and high PDI groups independently (S18 and S19 Tables). Solid lines and shaded regions reflect the mean and SEM of the weighted Bayesian model fits. The center inset shows the exponential fit of the distribution of PDI global scores from study 1 (grey line) and study 2 (black line), indicating the cutoffs for high and low PDI by vertical dashed lines. (c, d) Logit-belief updates as a function of logit prior by bead ratio for the low (c; S20 Table) and high (d; S21 Table) PDI groups. Group medians of individual medians for logit-belief updates are shown and other conventions follow Fig 3C. Solid lines and shaded regions reflect medians and 95% bootstrapped confidence intervals of the weighted Bayesian model fits. (e) Evidence asymmetry slopes are plotted against ω₁ by group. Other conventions as in Fig 3E. Marginal violin plots show the group medians and interquartile ranges. The asterisk indicates a significant rank-sum test comparing group medians of ω₁. (f) Mean final estimate differences are plotted against ω₁. The marginal violin plot shows the group medians and interquartile ranges. (e, f) The solid black line shows model predictions as in Fig 2E and 2F. As in Fig 3, after excluding outliers [52] (ω₁<0.82; 11 outliers), the correlation between ω₁ and the evidence asymmetry slope was still significant (ρ = -0.37, p = 1.08 x 10⁻⁴), as was the correlation between ω₁ and the mean final estimate difference (ρ = -0.47, p = 7.32 x 10⁻⁷).

https://doi.org/10.1371/journal.pcbi.1010796.g004

Direct replication of results from study 1.

As further validation of our task and model, we replicated the critical results from study 1 in the main sample of study 2 (Fig 4 and S2 Text), including the winning model (S6 Fig; S8 Table).

Group differences in sequential base-rate neglect reflect variability in real-world odd beliefs.

Having ensured enough variability in odd beliefs (i.e., PDI scores), we tested whether the high and low PDI groups differed on the relevant measures of sequential base-rate neglect. Both groups separately showed recency biases (mean final estimate difference and evidence asymmetry slope; all p<0.009), but only the high PDI group showed the prior-dependent belief-updating effect (low PDI: p = 0.41; high PDI: p = 0.02; sign-rank tests). There were no group differences for the recency bias measures (mean final estimate difference or evidence asymmetry slope; all p>0.48) or for the prior-dependent belief-updating effect, although the latter trended towards significance (p = 0.083, rank-sum test; Fig 5A). Crucially, the model-based measure of sequential base-rate neglect did differ between the groups, with more sequential base-rate neglect (lower ω₁) in the high PDI compared to the low PDI group (p = 0.018; rank-sum test; effect-size Cliff’s delta δ = 0.30). No group differences were observed in the other model parameters (: all 0.43>p>0.09; rank-sum tests; -0.10>δ>-0.21; Fig 5A and S22 Table).

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Fig 5. Prior underweighting relates to belief oddity outside the laboratory.

(a) Summary of group-level differences (S22 Table) for model-based and model-agnostic measures between the low (n = 57) and high (n = 34) PDI groups. Bar plots are effect size (Cliff’s delta, δ) and 95% confidence intervals. (b) Scatterplot showing a negative correlation between ranked mean PDI scores and ω₁ (n = 116). Values are adjusted by parameter values and the model fit (RMSE) for specificity as in the partial Spearman correlation analysis. Boxplots show medians (blue lines) and 25^th and 75^th percentiles (bottom and top edges, respectively). The solid black line reflects the least-squares linear fit to the data points. Mean PDI is the average of the global PDI scores across the pre-screening and the experimental sessions for each participant.

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Consistent with the observed group differences, an exploratory dimensional analysis (including 25 participants with PDI scores in an intermediate range between the high and low cutoffs in addition to the 91 comprising our primary groups per the pre-screening protocol; n = 116) showed that individuals with more unusual beliefs tended to exhibit lower ω₁ (ρ = −0.25, p = 0.007) and a trend towards stronger prior-dependent belief-updating (i.e., a more negative slope; ρ = −0.17, p = 0.065). The relationship with ω₁ held after controlling for all three parameters and the model RMSE (ρ = −0.22, p = 0.021; Fig 5C and S23 Table). The relationship with ω₁ also held after controlling for demographic variables including age, biological sex, race, education, handedness, smoking and drug use status, and previous hospitalizations for psychiatric and neurological conditions (ρ_partial = −0.24, p = 0.021), and none of these variables related to PDI scores. Our secondary measure of odd beliefs, the Paranoia Checklist, also correlated with ω₁ (ρ = −0.23, p = 0.014) and the prior-dependent belief-updating (ρ = −0.12, p = 0.039). Overall, the results of study 2 suggest that a laboratory measure of sequential base-rate neglect relates specifically to odd beliefs outside the laboratory.

Functional explanations for sequential base-rate neglect.

Thus far, we have shown evidence that human behavior in a sequential belief-updating task generally conforms to the predictions of a weighted Bayesian model of sequential base-rate neglect. Specifically, this model jointly predicts a recency bias and a pattern of prior-dependent updating as well as interindividual relationships with prior underweighting that we observed empirically. Further, interindividual variability in sequential base-rate neglect correlates with real-world belief oddity. However, this descriptive model does not provide a normative explanation as to why sequential base-rate neglect is such a predominant feature or why it varies across individuals. It also does not address whether prior underweighting may or may not be an optimal strategy under realistic constraints.

Study 3 thus aimed to address these outstanding mechanistic questions of why people exhibit base-rate neglect and whether it could reflect an optimal strategy. To do so, we considered models that explain variation in prior weighting as a rational response to external or internal factors. We specifically considered a first class of functional models that explains sequential base-rate neglect and its variability as a consequence of perceived variability in the environment [45,46] and a second class that explains it as a rational adjustment to a noisy internal sampling process [16]. In the first, in a volatile environment where the underlying evidence-generating process can change abruptly, the relevance of evidence before an inferred change point should be proportional to the certainty that a change point occurred. Thus, an optimal agent that perceives the environment as volatile will tend to decrease the prior weight around potential change points, therefore exhibiting sequential base-rate neglect. This class of model should be less applicable to the stable environment in the current task, but we reasoned that participants could still assume some degree of volatility despite explicit instructions to the contrary. The second class of models prescribes the optimal behavior for an agent with limited cognitive resources [57]. Under the noisy-sampling model [16] within this class, the agent can only access an imprecise, noisy representation of prior beliefs through random sampling of its internal representation (i.e., the distribution of the logit prior resulting from additive noise); it is possible to increase precision of the prior representation by increasing samples at the cost of allocating more internal cognitive resources, but the optimal strategy balances this cost against that of prediction inaccuracy (Fig 6A). Given this, the optimal strategy in this capacity-limited agent consists of decreasing the prior weight more in response to greater noise in the prior representation [16]. Because more noise in the prior representation should lead to more variable responses, even after accounting for structured variability due to sequential effects the noisy-sampling model predicts a correlation between the degree of sequential base-rate neglect and (unstructured) response variance. In contrast, the alternative volatility-based model we considered here does not predict this correlation in the context of our task (S7 Fig).

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Fig 6. Relationship between prior underweighting, prior noise, and response variance.

(a) Visual Schematic of Noisy Sampling Model. The noisy-sampling model captures an iterative sequential belief updating process where the internal representation of prior (and likelihood information) is noisy (see Methods). This is based on an agents’ uncertainty about the true values of the prior and likelihood, given recent evidence, with variances and , and their assumed distributions of priors and likelihoods, with variances and . Note that variables are in logit space and noise consists of an additive zero-mean Gaussian distribution. Critical to the model are a noisy representation of the prior and likelihood where the noise is given by the functions and . More noise (e.g., due to higher ) leads to more random variability in responses reflecting the posterior belief (even for repetitions of identical sequence fragments, as captured by the model-agnostic measure of response variance). Optimal inference results from adjusting weighting commensurate with the degree of noise, with optimal weights given by the functions and . Finally, the optimal posterior is a weighted sum of the noisy prior and noisy likelihood in logit space. Model fitting used 4 free parameters, 1 shared parameter and condition-specific parameters (3), and a grid search with 4 fixed parameters for (1) and (3) (Methods). (b) Scatterplot of ranked response variance rank and ω₁ showing a negative relationship indicating that individuals with more sequential base-rate neglect have more variability in their probability estimates for identical sequence fragments (Methods). (c) Scatterplot of ranked response variance and prior noise showing a positive relationship indicating that the model-agnostic measure of response variability scales with the model-derived measure of prior noise. (b, c) Boxplots reflect median (blue) and 25^th and 75^th percentiles (bottom and top edges, respectively). Black lines show the least-squares linear fit of the data points. (e, f) Noise-corrupted parameter recovery analysis for the weighted Bayesian model (e) and the noisy-sampling model (f). The y-axis shows the percent deviation in the recovered versus the original parameter values. The x-axis shows the magnitude of the late Gaussian noise added at the response level in the model simulations in standard deviation. Each grey line depicts a single agent defined by a set of parameter values across a range of noise levels. The red shaded area indicates the estimated range of response variance found in the actual data as a 95% confidence interval based on the median response variance (see Methods). On average (black line), the critical parameters are adequately recovered, without systematic biases in their estimation for meaningful levels of late Gaussian noise (particularly for the weighted Bayesian model), indicating that low-level factors such as general inattention or random responding are unlikely to explain variability in ω₁ values.

https://doi.org/10.1371/journal.pcbi.1010796.g006

We thus assessed the interindividual correlation between prior-weight ω₁ and response variance across all 267 participants from studies 1 and 2. A clear correlation was observed with ω₁ when using the unexplained variance by the weighted Bayesian model (the model RMSE) as an index of unstructured response variability (ρ = −0.40, p = 8.9 x 10⁻¹², Spearman correlation; Fig 6A). This relationship was also present in each independent sample (S24 Table). To circumvent potential artifacts of modeling, we also derived a model-agnostic measure of response variance focused on the unstructured variability of responses under identical circumstances—specifically, the aggregate response variance of logit probability estimates for repeated, identical sequence fragments matched on bead color and bead ratio (Methods), which we refer to as the response variance for simplicity and which captures variability that cannot be attributed to sequential evidence-order effects. Using this measure, we again found a correlation with ω₁ in the expected direction (ρ = −0.46, p<2.22 x 10⁻³⁰⁸; Fig 6B). Again, this relationship was also present in each independent sample (S24 Table). Although this result does not rule out the broader class of volatility models, it is more consistent with the noisy-sampling model; we thus further explored the ability of the latter model to capture our data.

The noisy-sampling model captures belief-updating behaviors described by the weighted Bayesian model.

The noisy-sampling model posits that the prior and likelihood weights of the weighted Bayesian model (ω₁ and ω_2(l)) scale negatively with the respective noise in the representation of the prior and likelihood, captured respectively by parameters and (S25 Table). The noisy-sampling model also includes parameters and that reflect the uncertainty in the distribution of logit priors and logit likelihoods that the agent might encounter (here held constant for model fitting to avoid parameter tradeoff; Methods). Perhaps unsurprisingly given that the structure of the noisy-sampling model reduces to the weighted Bayesian model, when fitted to our data (Methods) the noisy-sampling model captured comparable variance (correlation of explained R² between models: ρ = 0.93) and the σ² noise parameters closely correlated with the corresponding weights of the weighted Bayesian model (mean Spearman correlation ρ = −0.92; Fig 6C and S8 Fig) in the full sample combining studies 1 and 2 (n = 267). Under the noisy-sampling model, behavioral variability is partly due to noise in the internal representation of variables such as the prior. If this is true and it explains the observed correlation between ω₁ and response variance, prior noise should correlate with response variance. Consistent with this, the fitted parameter correlated with response variance (ρ = 0.35, p = 5.06 x 10⁻⁹; Fig 6D and S24 Table). Control analyses evaluating contributions of suggested that this parameter had no meaningful contribution to response variance or base-rate neglect (S9 Fig and Methods).

Alternative explanations to noisy sampling.

A possible alternative explanation of the observed correlation between prior weight ω₁ and response variance may be that individuals respond more inconsistently not because of noisy internal representations but due to other lower-level factors such as distraction or late motor noise. In other words, some inattentive participants could in principle tend to respond randomly. Although this is unlikely based on control analyses (S8 Fig and S9 Fig), if this were the case, perhaps data from these individuals was better fitted with lower ω₁ values due to modeling artifacts. To evaluate this possibility, we assessed robustness of parameter recovery for the weighted Bayesian model and the noisy-sampling model in the presence of levels of late noise that could capture random responding during the task (Methods). These analyses showed that parameter recovery of the relevant parameters (ω₁ and ) had no appreciable biases at levels of late noise matching the observed behavioral variability in the data (Fig 6E and 6F), suggesting that variability in their fitted values is unlikely to stem from lower-level factors irrelevant to the noisy-sampling model. Moreover, an explanation of prior underweighting in terms of inattentiveness may predict modulations of response times by ω₁ that were not present in the data (S27 Table). Altogether, these results speak against an explanation in terms of inattention or random responding and support noisy representation of prior beliefs as a more tenable explanation for sequential base-rate neglect.

Relationship between prior noise and real-world odd beliefs.

Because belief oddity correlated with sequential base-rate neglect (lower ω₁) in study 2, and the previous results imply that prior noise () could explain sequential base-rate neglect, we next asked whether prior noise could account for belief oddity. In the main sample from study 2, the high PDI group showed higher than the low PDI group (p = 0.007, rank-sum test; δ = -0.34; Fig 7A). No group differences were observed in the other model parameters (: all 0.40>p>0.12, rank sum tests; 0.18>δ>-0.53) or in response variance (p = 0.12, rank-sum test; δ = -0.20; S26 Table). These results suggest that high PDI may be specifically associated with increased prior noise.

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Fig 7. Prior noise relates to belief oddity outside the laboratory.

(a) Summary of group-level differences for noisy-sampling model-based and model-agnostic measures between the low (n = 57) and high (n = 34) PDI groups. Bar plots are effect size (Cliff’s delta, δ) and 95% confidence intervals. (b) Scatterplot showing a positive correlation between ranked mean PDI scores and (n = 116). Values are adjusted by parameter values and the model fit (RMSE) for specificity as in the partial Spearman correlation analysis. Boxplots show medians (blue lines) and 25^th and 75^th percentiles (bottom and top edges, respectively). The solid black line reflects the least-squares linear fit to the data points. Mean PDI is the average of the global PDI scores across the pre-screening and the experimental sessions for each participant.

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An exploratory dimensional analysis (using the same sample as in Fig 5B and 5C) further showed a correlation between prior noise and more unusual beliefs (ρ = 0.29, p = 0.002; Fig 7B), even after controlling for all three parameters and the noisy sampling model RMSE (ρ = 0.265, p = 0.0048; S24 Table). Altogether, these results suggest that noisy prior representations may explain sequential base-rate neglect and interindividual variability in odd beliefs outside the laboratory.

Task aim.

We developed a modified beads task building from previous work[31] where participants had to infer the identity of a hidden state (blue or green box) based on multiple samples of evidence (colored beads). We elicited probability estimates about the identity of the hidden state after each sample of evidence, allowing us to track the development of beliefs over time. We manipulated both the strength of the evidence (bead ratio in the hidden box) and, critically, the order in which the evidence samples were presented. The same task was used in studies 1, 2, and 3.

Trial structure.

Trials started with a 3-s presentation of two boxes with the same majority-to-minority bead ratio and different majority bead color (i.e., the blue box and the green box). To enhance clarity, the border of each box indicated the color of the majority bead color in the box and the contents of each box were displayed in text above each box (e.g., “60 blue, 40 green”; Fig 1A). One box was presented on the left side of the screen and the other on the right side of the screen. Next, participants were shown a white box with a black border and a question mark, which represented the hidden box. The first time it was displayed in a trial, participants just saw this hidden box for 1.5 s. On subsequent presentations of the hidden box, an animated green or blue bead rose up out of the box with this animation lasting 1.5 s. Participants then reported a probability estimate about how likely they thought the hidden box was the blue box or the green box. The top half of the screen showed a visual record of all beads shown so far within the trial, so as to minimize the working-memory burden and associated interindividual variability. The lower half of the screen displayed a black slider bar used to submit the probability estimate. Percentage values above each extreme of the slider indicated the complementary probability estimates for each box. The slider tick did not appear until a participant moved the mouse, and its starting point was randomized after each bead draw to minimize anchoring. Probability-estimate responses were self-paced and the response window was unlimited. After 8 samples were drawn and 9 probability estimates submitted, a binary choice for the hidden box was prompted. On this choice screen, the boxes were labeled “Left” and “Right” and participants had to respond with a corresponding (left or right) button press within an unlimited response window. When a response was submitted, the border of the selected box changed to yellow for 0.25 s to provide feedback that the selection was recorded. A blank screen was then presented for 0.5 s before the next trial began.

Task structure.

Participants completed 55 trials of the probability estimates task. At the beginning the experiment, participants were instructed that one of two boxes was randomly selected and hidden with equal probability: one containing mostly blue beads (blue box) or one containing mostly green beads (green box). One box was presented on the left side of the screen and the other on the right. The location of each box was determined at random on each trial. For a given trial, the bead ratio could be 51:49, 60:40, or 90:10, with each box displaying reciprocal ratios of bead colors. The participants’ task was to identify which box was selected and accurately estimate its probability. During each trial, 8 bead samples were presented, one at a time, and probability estimates were prompted before the first bead and after each of the beads about the probability that the hidden box was the blue or the green one. Participants were told that the individual beads were drawn randomly with replacement. To endow the estimates with instrumental value, after seeing 8 beads participants made a binary choice about the identity of the hidden box.

Incentive compatibility.

During the instructions, to incentivize responses that accurately reflected true beliefs and preferences, participants were informed they would be given an endowment of $10 that they could keep in its entirety (losing $0 or $5) based on their performance. After they completed all blocks of the experiment, $0 or $5 were subtracted from the endowment based on their performance on one randomly selected response. This could be a probability estimate (out of the 9 per trial over all trials) or a binary box choice (1 per trial). To determine the payoff, we instituted a binarized scoring rule [39] that is more robust to risk preferences and produces more accurate estimates than other commonly used methods [39], particularly when combined with potential loss from an endowment as in our implementation and in previous work leveraging endowment effects [101,102] to maximize task engagement and accuracy. If a probability estimate was chosen to determine the payoff, the probability of losing $5 was a function of the squared error of the reported probability estimate relative to the objective probability [103]. Specifically, a random value k from 0 to 1 was selected and participants lost $5 if the squared error of their chosen estimate was larger than k or $0 otherwise. The binarized scoring rule thus implies a quadratic loss function where the probability of losing $0 or $5, rather than the loss magnitude, depends on the precision of reported probability estimates. This leads to a U-shaped relationship between the expected value of a response and the posterior probability (S13 Fig). We also applied the binarized scoring rule to box choices, which in this case reduced to losing $5 when the choice was incorrect or $0 otherwise.

At the end of the task, participants were shown the selected response and the payout realization as explained above. To make the underlying principle of the scoring rule clear, an accessible explanation without excessive mathematical detail and several examples were presented to participants during the instructions (S1 Movie). To ensure comprehension, four of the miscomprehension quiz questions (S31 Table, Questions 1, 3, 4, and 9) specifically probed participants’ understanding of the scoring rule.

Instructions, practice, and comprehension checks.

To ensure participants completed the task within a reasonable time frame and in one session, they were required to complete the entire experiment within 4 hours (S14 Fig). The MTurk advertisement indicated that the task could take up to 2 hours to incentivize participants to minimize breaks. To minimize the incentive to rush through the task, participants were required to perform task trials for at least 40 minutes (and received additional trials if they completed the actual experimental trials earlier). To ensure task comprehension, participants were given comprehensive and detailed instructions for ~15–20 minutes. After the instructions, participants were required to complete a miscomprehension quiz (S31 Table). They were required to achieve 100% accuracy on the quiz or retake it until they did, consistent with prior work [48]. After the quiz, participants completed 3 practice trials, one with each possible bead ratio, and could repeat the practice if they wished. The practice-trial sequences were not used in the main experiment. A video demonstrating the instructions, quiz, and practice trials is available (S1 Movie).

Sequences of evidence.

Bead sequences were defined by the specific order of majority-to-minority beads. The color (blue or green) of the majority beads in the hidden box, which determined its identity, was randomly determined on each trial. Each bead-ratio condition comprised a different set of pre-determined fixed sequences; these were chosen from a broader set of all possible sequences of beads drawn randomly with replacement, in line with the instructions. Out of the 55 trials, there were 26 unique sequences of evidence order (S30 Table). Of those, 16 sequences were identical across the 60:40 and 90:10 conditions (“matched trials”). Sequences were presented in blocks of 11 trials, organized by the bead-ratio condition. The order of blocks was the same for each participant: 60:40, 90:10, 51:49, 60:40, 90:10. Within each block, sequences were selected at random without replacement from the sequence set. We selected sets of sequences for which the distribution of majority beads over trials for a given bead ratio matched the distribution of expected sequences of that ratio. To achieve this, 4 sequences were unique to the 51:49 condition, 6 sequences to the 60:40 condition, and none to the 90:10 condition.

Critically, we constructed mirror-opposite sequence pairs to facilitate isolation of sequence-order effects. Further, we aimed to vary sequences in their degree of evidence asymmetry, or how extremely front- or back-loaded the majority beads were in a sequence. Here, and throughout the manuscript, sequences are presented such that 1 (or black) represents the majority bead color, and 0 (or white) the minority color. We quantified evidence asymmetry as a linear weighted sum of a binary sequence, for instance [1 1 0 1 1 1 1 0], with each element in the sequence vector weighted as a function of their linear distance from the middle (weights: [-3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5]). The result of the weighted sum (in this example, -2) thus indicated that majority beads were presented mostly towards the beginning of the sequence (front-loaded) with negative values. Positive values would indicate that majority beads were presented mostly towards the end of the sequence (back-loaded). Greater absolute values indicate more extreme back- or front-loading. Mirror-opposite sequence pairs had identical bead-ratio, number of majority beads, and absolute evidence asymmetry such that their comparison would isolate sequence-order effects. In particular, recency biases should manifest as more certain beliefs (favoring the true hidden box) for back-loaded (compared to front-loaded) sequences, particularly in sequences with greater evidence asymmetry.

Overall, we selected trials to span a range of evidence asymmetry, bead-ratio conditions, and total number of majority beads (Fig 1B).

Study 1.

Before completing the probability estimates beads task, participants completed a demographic survey (S1 Table) and the PDI²² (see S29 Table for the complete set of items). The PDI is a 21-item questionnaire that measures odd, delusion-like ideas in the general population. The experiences interrogated a range of more common experiences such as “do you ever feel as if some people are not what they seem to be?” or “are you worried that your partner may be unfaithful?” to more unusual ones, like “do you ever feel as if you are a robot or zombie without a will of your own?” or “do you feel as if things in magazines or on TV were written especially for you?” For each experience, the participant can endorse the belief with a Yes or No response. If they report No, then the global item score is 0. If they report Yes, they must then report on a scale of 1 to 5 how distressing the belief is (1 = not distressing at all, to 5 = very distressing), how often they think about it (1 = hardly ever, to 5 = all the time), and their conviction about it (1 = don’t believe it’s true, to 5 = believe it is absolutely true). The global item score is the sum of these three responses plus the “Yes” endorsement. The global PDI score (with a possible range of 0 to 336) is the sum of all the global item scores (each between 0 and 16).

Study 2.

Participants completed the PDI and the Paranoia Checklist[104]. Although our primary measure was PDI, we included the Paranoia Checklist for exploratory purposes to assess the generalizability of the results and to confirm that recency bias was generally related to odd beliefs and not to paranoid beliefs specifically. The Paranoia Checklist is a 18-item measure of paranoid beliefs in the general population.

Study 1.

Participants were recruited through Amazon MTurk, and the experiment was run on gorilla.sc [105]. They were paid $10 plus a performance bonus of $5 or $10. Using MTurk filters, we only invited participants who had already successfully completed at least 50 tasks with a 90% approval rate, who were under 55 years old, located in the US, and had an MTurk Masters Qualification (given to workers who “demonstrate a high degree of success in performing a wide range of [tasks] across a large number of requesters”).

The experiment comprised multiple components, including the task itself and questionnaires. Several participants began the study, completing the questionnaires but not the task. These non-completers were excluded from all analyses. Importantly, at least for those who completed the questionnaires (S1 Table and S2 Table) we did not find differences in belief oddity (p = 0.484) or evidence for selection biases in completers on most relevant measures.

We also implemented exclusion criteria based on performance. First, to avoid “bots”, we assessed if average responses were above a minimum of 350 ms (the approximate time needed to shift endogenous attention [106]). No participants were excluded by this criterion. Second, we limited our analysis to participants who identified the “correct” box at the end of the bead sequence with accuracy above 68% for 60:40 and 90:10 bead-ratio conditions based on the binomial chance level (15 correct trials out of 22; accuracy criterion). Third, to assess engagement in the task, we used a linear regression analysis predicting participants’ subjective probability estimates based on the random starting point of the cursor on the sliding scale (see task details below) and the optimal Bayesian estimate (i.e., the objective probability). Participants were excluded if the random cursor start position significantly predicted their subjective estimates and the Bayesian estimate did not. If both conditions were satisfied (random-estimates criterion), we reasoned that the participant was likely trying to the task as fast as possible with no regard for accuracy. If the optimal Bayesian estimate also predicted subjective estimates, we reasoned that the participant may have been engaged in the task but was anchoring to the random cursor-start location, which was insufficient for exclusion. A total of 213 participants began this study. 43 were non-completers, and 8 were excluded for meeting either the accuracy or the random-estimates criterion.

To further determine if participants were correctly engaging the task, we developed 2 heuristic models reflecting strategies that participants may have used and which would not reflect belief updating. The first heuristic model (no-prior model; equivalent to ω₁ = 0) reflects a strategy where participants report fixed belief certainty in favor of the most recent evidence sample. For example, for a blue bead they would report 0.8 in favor of the blue box and for a green bead 0.8 in favor of the green box. The second heuristic model (observed-proportion) reflects a similar strategy with the difference that the favored box is based on whichever color has been drawn more often. For instance, after observing 3 blue beads and 1 green bead the participant could report 0.8 in favor of the blue box and only change their estimate to 0.8 in favor of the green box after observing more green than blue beads. The heuristic models, along with all the belief updating models (S28 Table), were fit to the data for the 60:40 and 90:10 conditions. The 51:49 condition was not used here because the low evidence strength in this condition makes it harder to determine whether estimates are consistent with heuristic strategies. We excluded any participant whose 60:40 and 90:10 data was fit best by one of the heuristic models in a formal model comparison using the BIC [107] at the individual level. Based on this criterion, 5 participants were excluded for being best fit by the no-prior model, and 6 participants by the observed-proportion model. There were no significant differences in demographic characteristics between participants who were included in the study and those who were excluded based on performance criteria or non-completers (S2 Table).

In sum, 237 participants were recruited from Amazon MTurk, of whom 170 completed the task. Of these, 8 were excluded for poor accuracy or random responding and 11 because their data was best fit by a heuristic model which suggested they did not engage the task as intended. After exclusions, 151 participants were retained and included in the primary analysis. S1 Table shows the demographic information of all 151 completers who were included in the analysis for study 1. Quality checks indicated the data was of comparable quality than similar in-person studies (S14 Fig and S15 Fig; see Online Data Quality in Methods).

Study 2.

Participants were recruited through Amazon MTurk, and the experiment was run on gorilla.sc [105]. We implemented the same MTurk filtering criteria as in study 1, with the exception that we did not limit participants to MTurk Masters so as to increase participation. We also excluded anyone who already participated in study 1. Study 2 consisted of two parts. For part 1, participants were paid $2 to complete 2 questionnaires. 547 participants started part 1, and 512 completed it (93.6%). Participants were invited back for part 2 based on their questionnaire scores from part 1. 241 participants were invited to participate in part 2. For part 2, participants completed 1 questionnaire and the probability estimates beads task. The task and incentive structure was identical to study 1. 213 participants started part 2, and 143 participants completed it (67.14%). We applied the same performance and model-based exclusion criteria as in study 1. Using these criteria, 10 participants were excluded based on their performance (accuracy and random-estimates criteria) and 17 due to evidence (BIC) favoring heuristic models. There were no significant differences in demographic characteristics between participants included in the study and those excluded based on poor performance (S3 Table). For study 2, we analyzed the data for 116 participants (S1 Table). Across all 116 retained participants, PDI scores were stable between the pre-screening and experimental sessions (ρ = 0.97) and both correlated strongly with the secondary measure of odd beliefs, the Paranoia Checklist (all ρ>0.79). Quality checks again suggested comparable quality to similar in-person studies (S14 Fig and S15 Fig).

Pre-Screening and PDI Classification.

To ensure a wide enough range of odd beliefs and sufficient sampling of meaningfully high levels [54], study 2 pre-screened participants based on the PDI²². Participants with high or low belief oddity based on their PDI scores were invited for the experimental session, with the cutoffs based on reported norms for PDI global scores⁴⁷ (mean of 58.9 and standard deviation of 48.0 in healthy individuals): mean plus 0.5 standard deviation (>82.9) for the high PDI group and mean minus 0.5 standard deviation (<34.9) for the low PDI group. For secondary analyses, we also invited participants with high (>17.15) or low (<6.65) frequency scores on reported norms for the Paranoia Checklist⁹². Participants who were invited solely based on the Paranoia Checklist scores were only included in exploratory dimensional analyses. Participants in the high and low PDI groups were gender- and age-matched (within 2 years). Those who completed the experimental session completed the PDI a second time (typically within 1–2 days of the pre-screening) and the mean PDI across both sessions was used for dimensional analyses.

Data from these participants was again high quality (S9 and S10 Figs).

Study 3.

Study 3 re-analyzes combined data from studies 1 (n = 151) and 2 (n = 116) in 267 participants.

Weighted Bayesian belief-updating model and variants.

We fit several weighted Bayesian belief updating models to the draw-by-draw probability estimates for each participant individually and extracted best-fitting parameters for each model. All models in the model comparison were variants of a weighted belief-updating model: logit(posterior) = ω₁∙logit(prior)+ω₂∙logit(likelihood).

In this model, logit(prior) represents the log-odds of the prior probability or belief on the current draw before integrating the likelihood, and it is equivalent to the posterior probability after the previous draw. logit(likelihood) represents the log-odds of the likelihood (or the log-likelihood ratio), which is the strength of the sensory evidence given by the bead-ratio for a specific bead draw with respect to the correct box. logit(posterior) represents the updated log-posterior ratio about the probability that the beads are coming from the green or blue box after combining the prior and the likelihood terms. The prior-weight ω₁ is a free parameter that acts as a multiplicative weight on the prior belief; it affects how much older evidence is incorporated into the updated beliefs, controlling a primacy-recency bias. Prior underweighting (ω₁ < 1) captures sequential base-rate neglect, limiting belief certainty (Fig 2C) and inducing a recency bias (Fig 2D and 2E) [2,4,17]. The likelihood-weight ω₂ is a free parameter that scales the likelihood term multiplicatively and equally for older and newer samples of evidence, producing distinct effects from the prior-weight ω₁ (S2D Fig).

Model fitting was performed for each subject using the Matlab function fmincon [118] in order to minimize the root mean squared error (RMSE) between the model-estimated probabilities and the probability estimates reported by the participant. Only estimates after bead draws were used for fitting, and the participant’s first estimate before the first bead draw defined the starting prior belief for a given trial. Data for sequences associated with an incorrect final decision were excluded from analyses. For robustness, participants’ data were each fit 100 times to each model, using random starting points between 0 and 20 for each free parameter. Bounds were set to 0 and 20. The parameters associated with the iteration yielding the lowest RMSE were taken as the best-fitting parameters for the participant and model. Formal model comparison (for the same 10 models used for comparison in our previous work [31]) was conducted based on the Schwarz Bayesian Information Criterion (BIC)[107]: , where n is the total number of fitted probability estimates (per participant), error is the difference between the actual probability estimates and the simulated probability estimates, and l is the number of parameters in the model. BIC values were used to calculate the protected exceedance probability (using the Variational Bayes Toolbox [119]; Fig 3D and S6 Fig) for group-level Bayesian model selection.

Noisy-sampling model.

Azeredo da Silveira and Woodford [16] described a noisy-sampling model of belief updating where agents do not have access to the full prior distribution and instead represent prior beliefs imprecisely via noisy internal samples. Under this model, a rational response to imprecision in prior beliefs given the costs of precision and prediction inaccuracy is to underweight prior beliefs. This model thus provides a functional account for sequential base-rate neglect: lower prior-weight results from, and is inversely proportional to, noise in prior beliefs.

Here, we specify this model in the context of sequential belief-updating in our task. The noisy log-odds of the prior with respect to the true underlying state of the hidden box is:

r_prior reflects the noisy internal representation of the prior. π_majority reflects the prior probability in favor of the true underlying state of the hidden box. π_minority reflects the prior probability in favor of the incorrect state of the hidden box, where π_majority+π_minority = 1. v_p reflects the Gaussian noise (in logit space) of the internal representation of the prior, which is centered around 0 and has a variance of . Similarly, the noisy log-odds of the likelihood with respect to the true underlying state of the hidden box is:

r_likelihood reflects the noisy internal representation of the likelihood. λ_majority reflects the likelihood in probability space in favor of the correct state of the hidden box, and. λ_minority reflects the likelihood in favor of the incorrect state, where λ_majority+λ_minority = 1. v_l reflects the Gaussian noise (in logit space) of the internal representation of the likelihood. The Gaussian noise is centered around zero and its variance may vary per bead-ratio condition, where can take on values , or depending on the condition.

To calculate an optimal estimate of a participants’ beliefs in response to new evidence, we must also define a probability distribution over the possible true underlying states; that is, we must define the prior distributions from which the values {π, λ} may have been drawn. Here we define these distributions as centered around their corresponding probability ratio, where the variance of the distribution is given by or . Further, may similarly take on different values , or per condition. The complete generative model over true possible situations and the participants’ noisy internal representations is specified by eight parameters: .

Conditional on the prior (before the observation of a new bead draw), we can determine what optimal Bayesian inference of the true underlying state of the hidden box would be. The equations described above imply that the true log-odds (based on previously available information), , and the prior have a joint, bivariate, Gaussian distribution. Consequently, the distribution of z_prior conditional on the noisy representation of the prior, r_prior, will also be a Gaussian distribution:

From this, the implied probability that the true state of the hidden box is given by: where f(z_prior|r_prior) is the density function of the conditional distribution. In order to compute this quantity as a function of the ω² and σ² parameters, we use an analytical approximation[120] of this integral:

ρ_prior is a correction owing to the fact that the posterior distribution is not concentrated entirely at its mean. Then, we can substitute in the formula for prior to get:

Here, we show the calculation for the prior, but the calculation of the likelihoods follows the same logic and can be obtained by substituting the corresponding ω² and σ² parameters. To calculate the posterior after each bead draw, we simply add or subtract the likelihood from the prior depending on if the bead is in favor of or against the true underlying state of the hidden box. If the signal is in favor of the true underlying state of the hidden box, the posterior would be:

If the signal is against the true underlying state of the hidden box, the posterior would be:

In either case, ϵ = ϵ_prior + .

The ω₁ and ω_2(l) parameters in the weighted Bayesian model correspond respectively to the weights and in the noisy-sampling model and are thus inversely proportional to the noise parameters, , respectively. They are also inversely proportional to the parameters and representing the assumed uncertainty in the underlying logit prior and likelihood distributions. In visual schematic of the model in Fig 6, the prior weight is framed as the function , the likelihood weight as the function , and the noise parameters ϵ as the functions , and .

In fitting the model, consistent with previous work [121] we assumed that participants would adapt to the context of the task, acquiring a realistic estimate of the uncertainty underlying logit prior and likelihood distributions. Under this assumption and should be relatively constant across participants, and the primary source of interindividual variability should be reflected in the parameters. To avoid the possibility of parameter trade-off, our primary analysis fixed the and the 3 across the entire sample, but allowed the and the 3 parameters to vary freely. To determine the appropriate values for the and the 3 parameters, we conducted a 4-dimensional grid search of ω² values from 0 to 1.4 in steps of 0.2, fitting the 4 σ² parameters to each participant’s data for a given set of ω² parameter values. To do this we used the Matlab function fmincon [118] in order to minimize the RMSE between the model-estimated probabilities and the probability estimates reported by the participant. We then calculated the group-level BIC (based on RMSE across all trials and participants) and selected the ω² parameter values from the model with the lowest value; these were: , . Fitted σ² parameter values from the noisy-sampling model with these fixed ω² values were taken as best-fitting values and used in our main analyses. Since increased and parameter values could both lead to decreased prior weighting and response variance under this model (S7 Fig), we empirically assessed the possibility that , and not , could drive interindividual variability in base-rate neglect and response variance. Instead of using fixed parameter values, we took the best-fitting values for each individual. Critically, these individually best-fitting values were uncorrelated with the prior-weight ω₁ from the weighted Bayesian model and our measure of response variance (S18 Fig). Furthermore, comparisons of parameter values fitted for relevant subgroups (median-split groups based on ω₁ or response variance) were inconsistent with an alternative explanation of base-rate neglect in terms of variability in .

Parameter recovery analysis.

To generate simulated agents for parameter recovery, we sampled agent model parameters from the range of fitted parameters values found in the real data. Specifically, we randomly sampled parameters uniformly from the 10^th to 90^th percentile of values to limit the influence of extreme values. Responses were then simulated on the experimental trials that participants observed. Simulated observers started each trial with unbiased prior beliefs about the hidden box and posterior beliefs after each draw were updated in logit space according to the model. To evaluate the robustness of model fitting procedures to late (e.g., motor) noise, varying magnitudes of zero-mean Gaussian noise were added to the logit posterior beliefs after updating. This late noise was unrelated to the inference process, and thus only affected the agents’ reported noisy estimates and did not propagate to subsequent prior beliefs. To simulate realistic levels of late Gaussian noise, we estimated the variance that matched the variability observed in the data. First, we determined the 95% confidence interval of the median response variance at the group level in the actual data via bootstrapping. Next, we simulated ten sets of random agents (n = 267 per set, as in the combined dataset for study 3) across a range of late-noise variance levels. For each level, we calculated the median response variance at the group level and the mean of the medians across the sets. We determined the estimated noise range of the actual data to correspond to noise levels where this mean of medians overlapped with the 95% confidence interval of the median response variance in the actual data.