The effects of base rate neglect on sequential belief updating and real-world beliefs
Fig 2
Model predictions for sequential base-rate neglect under the weighted Bayesian model.
An agent with sequential base-rate neglect (ω1 < 1.00; for these simulations: ω1 = 0.88), in blue, is compared with a Bayesian ideal observer (ω1 = 1), in grey, on the 60:40 bead-ratio condition. Values of ω2(60:40) are 0.51 (or between 0.31 and 0.66 in the shaded regions in panels d-e) consistent with observed mean values (and 25th to 75th percentile range) in our prior work with a similar beads task [31]. (a) Simulated sequential probability estimates for two mirror-opposite sequences for a base-rate neglecting agent (blue/solid) and the ideal Bayesian observer (grey/dashed). Majority beads are shown as black and minority beads as white for illustrative purposes. Belief trajectories for front-loaded sequences show a gradient from dark to light and those for the back-loaded sequences transition from light to dark (b) Simulation of the recency bias, defined as the difference between the final probability estimate after 8 beads between mirror-opposite pairs, as a function of the absolute evidence asymmetry of the pairs. As in Fig 1, larger circles reflect sequences with more majority beads. The fit line shows the fixed effect of absolute evidence asymmetry on the final estimate difference. The simulated base-rate neglecter shows higher estimates for back-loaded sequences (compared to their front-loaded mirror opposites), particularly for sequence pairs with more evidence asymmetry. This effect varies with evidence strength and is strongest in the 90:10 condition (S2A Fig). See S2E Fig for a simulation of an agent that overweights the prior. The lower-case delta shows the example from (a). (c) Simulation of the magnitude of logit-belief updates as a function of the prior with respect to the color of the current evidence. For illustrative purposes, the x-axis has been discretized into bins equivalent to 0.1 increments of prior beliefs in probability space. The y-axis represents the mean magnitude of the logit belief updates (the difference in the log-odds of the prior and the posterior belief). The Bayesian ideal observer has constant logit belief-updates. In contrast, the simulated base-rate neglecter shows logit-belief updates that depend upon the prior belief, with relatively larger updates for prior-inconsistent evidence (left of the vertical dashed line) and smaller for prior-consistent evidence (right) (see S2B Fig for a condition-wise simulation). The fit line reflects the fixed effect of logit-prior on the logit-belief update, which we refer to as prior-dependent belief updating. The model predicts main effects of logit-prior and bead-ratio condition, but no interaction S2B Fig. See S2D Fig for a simulation illustrating the distinct scaling effects of ω2 and S2F Fig for a simulation of an agent that overweights the prior. (d) Simulation demonstrating the predicted relationship between ω1 and the evidence asymmetry slope (blue fit line from b). (e) Simulation of the predicted relationship between ω1 and mean final estimate difference (average of blue data points in b). (d,e) The blue and grey dots show the values for the base-rate neglecting and Bayesian ideal observers simulated in (a,b,c).