Fig 1.
Different environmental evidence weights cause decision biases.
a-b. Schematic of the Jar-Discrimination Task. Balls were drawn with replacement from one of two equally probable jars with different ratios of red to blue balls. Here h± denotes the probability that a red ball is drawn from the high (h+) and low (h−) jar. We consider conditions with symmetric priors and symmetric evidence (h− = 1 − h+; a), in which the red/blue ball observations had equal weights but opposite signs, or asymmetric evidence (h− ≠ 1 − h+; b), in which rare (in this example red) balls were weighted more heavily in a decision. c-d. The corresponding probability distribution of a 10-ball sample for a given number of rare balls drawn from the high jar (h+, top) and low jar (h−, bottom) for the symmetric (c) and asymmetric (d) evidence cases. Colored bars presented on the top axis denote an ideal Bayesian observer’s jar choice resulting from the associated log likelihood ratio (LLR; an LLR of zero results in a random response). e-f. Example of a 10-ball sample and corresponding choices of a Bayesian observer with varying relative ball weights. e. Ideal ball weights for the symmetric environment produce even response fractions. f. Ideal asymmetric weights produce a choice asymmetry in favor of the low jar. Deviations from the ideal weights in either environment produce decision biases.
Fig 2.
Suboptimalities are reflected by the psychometric function.
a. Illustration of how suboptimalities, such as mistuned ball weights or biased priors, compensate for (overweighting, bias in favor of high jar) or accentuate (underweighting, bias in favor of low jar) choice asymmetry in environments with asymmetric evidence, whereas increases in variability (inclusion of noise and/or variance) have a small impact on choice asymmetry. b. Examples of how a psychometric function fit to data is modulated by suboptimalities. An increase in noise decreases the slope, and a bias results in a horizontal shift of the psychometric function. We define variance as the mean absolute error between the best–fit psychometric function and the data, representing systematic aspects of strategies unaccounted by the LLR. c. Schematized bias-variance space showing how suboptimal bias and variance shift an observer’s location in bias-variance space. Bias was bounded between [−10, 10] to mitigate overfitting due to outliers. Positive (negative) biases corresponded to more (fewer) low-jar selections.
Fig 3.
Human subjects displayed choice asymmetries that deviated from the ideal observer.
a. Accuracy for each subject (N = 198, grey circles) and sample-matched ideal observer responses (grey diamonds) for each block: Control (CT), Hard Asymmetric (HA), Hard Symmetric (HS), Easy Asymmetric (EA), Easy Symmetric (ES). Population bootstrapped means (1000 iterations) and 95% confidence intervals are shown in bold. Model and subject population accuracy was significantly above chance in all cases (0.5; p < 0.05). b. Low-jar response fractions displayed as in a. Filled markers denote a significant population shift away from the prior (0.5; p < 0.05). c-d. Example psychometric function (line) fit to a sample subject’s high-jar responses (dots) for the HA block (c) and EA block (d) across all sample lengths. e-f. Bias and variance for individual subjects (points) obtained from fits of the psychometric curves to data from HA blocks (e) and EA blocks (f). Bias was bounded between [−10, 10] to mitigate overfitting to outliers. Positive (negative) biases corresponded to more (fewer) low-jar selections.
Fig 4.
Subjects used Bayesian and heuristic strategies in asymmetric blocks.
a. Bayesian models. Differences between the Noisy Bayesian model and alternative Bayesian models are underlined. b. Heuristic models. See Methods and ‘Formal model comparison’ section for more model details. c. Log Bayes factors (log(BF)) for each subject-block, computed between each alternative model and the Noisy Bayesian model. log(BF)>0 favors the alternative model, with log(BF)>1 or <−1 (dashed lines) providing strong evidence in favor of a given model [8]. Black (grey) markers indicate that the listed alternative model is (is not) the most likely model (percentage of subjects whose most-likely model is identified by strong evidence: 36% for Noisy Bayesian, 42% for Set ρ, 32% for Prior, 90% for Variable Rare, 87% for Rare Ball, 82% for Guess). d. Subjects categorized by the model that best describes their responses for the Hard Asymmetric (HA) and Easy Asymmetric (EA) blocks. For both blocks, a majority of the subjects’ responses were best described by Bayesian models (55% in HA, 86% in EA), but with a relatively high percentage of heuristic strategies under the HA condition.
Fig 5.
Increased bias and variance in asymmetric blocks corresponded to Bayesian subject model fits with mistuned parameters and heuristic subject model fits, respectively.
a. Left: Hard Asymmetric (HA) and Easy Asymmetric (EA) block bias-variance plots from Fig 3E and 3F, color-coded according to each subject’s best-fitting model described in Fig 4D. Triangles denote median values for the bias-variance fits for: 1) Nearly Ideal subjects (best fit by “Noisy Bayesian Set ρ” model), 2) Mistuned Bayesian subjects (best fit by “Noisy Bayesian” or “Prior Bayesian” models), 3) Heuristic subjects (best fit by “Variable Rare”, “Rare Ball”, or “Guess” models). Mistuned Bayesian and Heuristic groups that significantly (not significantly) differ from the Nearly Ideal group are denoted by filled (open) triangles based on a Wilcoxon rank-sum test with p < 0.05. Right: Group bootstrapped means (1000 iterations) and 95% confidence intervals for low-jar responses. Statistically significant differences between groups (two-sided t-test with unequal variance, p < 0.05) are noted with an asterisk. b. Estimated subject bias obtained from best–fit psychometric functions compared with the maximum-likelihood estimate (MLE) of the rare-ball weight, ρ, for subjects best fit by the Noisy Bayesian model in asymmetric blocks (dots, EA-grey, HA-black). Regression lines are shown for group-blocks with significant correlations (Spearman correlations, p < 0.05). Vertical lines indicating the rare-ball weights used by the ideal observer for each asymmetric block and symmetric blocks (orange) are included for reference. c. Estimated subject bias from fit psychometric functions compared with the MLE of the response bias (Prior) for subjects best fit to the Prior Bayesian model in the asymmetric block (marker legend as in b). Negative values correspond to a bias in favor of the low jar.
Fig 6.
More complex but suboptimal human strategies exhibited more bias.
a. Mutual information (MI) between the number of rare balls in a sample (|ξ|), the sample length (n), and the response (r) for each subject and block. b. Accuracy versus MI (computed as bootstrapped means from 1000 iterations per subject) for the Hard Asymmetric (HA) and Easy Asymmetric (EA) blocks. Dots represent data from individual subjects, color coded by subject’s best-fitting model described in Fig 4D. Black line represents the accuracy bound (the maximum accuracy attainable by the idea observer for a fixed MI in the limit of many trials). The dashed horizontal lines indicate the accuracy bound for maximum MI values. Note that points could exceed the asymptotic accuracy bound because the number of trials for each subject was finite. Median values for the Nearly Ideal, Mistuned Bayesian and Heuristic subject groups are indicated with triangles. In each case, filled Mistuned Bayesian and Heuristic triangles denote statistically significant differences in MI from the nearly ideal group (p < 0.05) based on a Wilcoxon rank-sum test. Median values for all 3 groups showed increase in both accuracy and MI ranking from lowest (Heuristic), middle (Mistuned Bayesian), highest (Nearly Ideal). c- d. Relationship between estimated bias (c) and variance (d) from the fit psychometric function for each subject and MI, triangles represented as in b based on statistically significant differences in bias or variance. e. Algorithmic complexity for each model. Bayesian models shown as the mean algorithmic complexity across sample lengths.