Probability Theory Plus Noise: Descriptive Estimation and Inferential Judgment

Abstract

We describe a computational model of two central aspects of people's probabilistic reasoning: descriptive probability estimation and inferential probability judgment. This model assumes that people's reasoning follows standard frequentist probability theory, but it is subject to random noise. This random noise has a regressive effect in descriptive probability estimation, moving probability estimates away from normative probabilities and toward the center of the probability scale. This random noise has an anti-regressive effect in inferential judgement, however. These regressive and anti-regressive effects explain various reliable and systematic biases seen in people's descriptive probability estimation and inferential probability judgment. This model predicts that these contrary effects will tend to cancel out in tasks that involve both descriptive estimation and inferential judgement, leading to unbiased responses in those tasks. We test this model by applying it to one such task, described by Gallistel et al. ). Participants' median responses in this task were unbiased, agreeing with normative probability theory over the full range of responses. Our model captures the pattern of unbiased responses in this task, while simultaneously explaining systematic biases away from normatively correct probabilities seen in other tasks.

On the generality and cognitive basis of base-rate neglect

“…This model shares similarities with the “Probability theory plus noise” model by Costello and Watts (Costello & Watts, 2014, 2016, 2018). The main difference is that the model by Costello and Watts puts the noise on the input representations (before integration), while we put it on the decision variable (after integration) 3 .…”

“…The main difference is that the model by Costello and Watts puts the noise on the input representations (before integration), while we put it on the decision variable (after integration) 3 . Although it has been repeatedly proposed in the literature that base-rate neglect could be a side effect of random noise (Costello & Watts, 2018; Erev et al, 1994; Hilbert, 2012; Juslin et al, 1997), the hypothesis has rarely been subjected to empirical tests in experiments with cognitive modeling.…”

“…This resulting variability is not only believed to be a major source of errors in perception (e.g., Drugowitsch, Wyart, Devauchelle, & Koechlin, 2016; Stengard & Van Den Berg, 2019), but may potentially also explain, at least in part, many of the biases that have been found in cognition (Erev et al, 1994; Hilbert, 2012). Building on that research, Costello and Watts have proposed that people’s probability judgments follow the laws of probability theory but are affected by random noise (Costello & Watts, 2014, 2016, 2018). If decision noise is not taken into account in the medical diagnosis task, any participant with noise in their decision process will be classified as non-Bayesian, even if the participant was using a Bayesian strategy.…”

Individual Differences in the Perception of Probability

“…The inferred bias also depends on the statistical approach used: Both repulsive errors and conservatism can be identified within the same data set, depending on the statistics used to characterize bias [12]. In addition, judgments are subject to internal noise, which can interfere with the identification of biases [12][13][14].…”

Editors’ Introduction: Cognitive Modeling at ICCM: Advancing the State of the Art

“…In human probabilistic reasoning, we are known to have systematic biases. Costello and Watts () report on investigating two central aspects of probabilistic reasoning through the use a cognitive model of inferential probability judgment. Their general model of descriptive probability estimation, inferential probability judgment, and the interactions between these two processes made predictions about patterns of agreement and disagreement by people and probability theory that were well supported by experimental results.…”