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The effects of base rate neglect on sequential belief updating and real-world beliefs
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 25th and 75th 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.
doi: https://doi.org/10.1371/journal.pcbi.1010796.g006