The effects of base rate neglect on sequential belief updating and real-world beliefs
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 25th and 75th 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).