People’s conditional probability judgments follow probability theory (plus noise)

Abstract

A common view in current psychology is that people estimate probabilities using various 'heuristics' or rules of thumb that do not follow the normative rules of probability theory. We present a model where people estimate conditional probabilities such as P(A|B) (the probability of A given that B has occurred) by a process that follows standard frequentist probability theory but is subject to random noise. This model accounts for various results from previous studies of conditional probability judgment. This model predicts that people's conditional probability judgments will agree with a series of fundamental identities in probability theory whose form cancels the effect of noise, while deviating from probability theory in other expressions whose form does not allow such cancellation. Two experiments strongly confirm these predictions, with people's estimates on average agreeing with probability theory for the noise-cancelling identities, but deviating from probability theory (in just the way predicted by the model) for other identities. This new model subsumes an earlier model of unconditional or 'direct' probability judgment which explains a number of systematic biases seen in direct probability judgment (Costello & Watts, 2014). This model may thus provide a fully general account of the mechanisms by which people estimate probabilities.

The Bayesian sampler: Generic Bayesian inference causes incoherence in human probability judgments.

“…Here we provided a generic Bayesian account of how this problem can be addressed. It turns out, unexpectedly, that the approach perfectly mimics the predictions, in expectation, for many judgments from a major recent theoretical account with strong empirical corroboration: the Probability Theory plus Noise (PT+N) model (Costello & Watts, 2014, 2016a, 2019. The general approach outlined here (whether using the Bayesian sampler or PT+N) also captures a variety of interesting further phenomena.…”

“…By contrast, the Bayesian sampler assumes that conditional probabilities are treated the same as any other kind of probability, and because only one variable needs to be checked when evaluating the samples, we make the simplifying assumption that the same number of samples, N , is drawn as for simple events. Therefore, the average predicted conditional probabilities of the Bayesian sampler are the same as those for simple events, which differs from the predictions of PT+N: However, when these conditions do not hold, the PT+N and Bayesian sampler do make distinguishable predictions for the probabilistic identities in which conditional probabilities are involved (see Table 1: fromẐ 9 toẐ 18 ) In particular, even if ∆d > 0, the PT+N model predicts that the expected values ofẐ 9 toẐ 13 should be strictly equal to 0 (Costello & Watts, 2016a), whereas the Bayesian sampling model predicts that these values can be different from 0. Past empirical work has shown that for a range of events these identities are very close to 0, but the pairs of events were not chosen to distinguish the two models.…”

“…However, Costello and Watts also derive a number of other identities that should not be preserved in the PT+N account. The predictions from PT+N of both the identities that were expected to match probability theory and those that were expected to deviate from probability theory were verified in a series of experiments (Costello & Watts, 2014, 2016a The PT+N model, at first glance, looks like a rival to a rational Bayesian account because it departs from rationality in the light of putative mechanistic factors, concerning the noisiness of memory. As we shall see, though, it turns out that a natural Bayesian sampling model generates predictions for a wide range of judgments that are, in expectation, identical to those of the PT+N model.…”

“…The Bayesian sampler trades off the coherence of probabilistic judgments for improved accuracy, and provides a single framework for explaining phenomena associated with diverse biases and heuristics such as conservatism and the conjunction fallacy. The approach turns out to provide a rational reinterpretation of "noise" in an important recent model of probability judgment, the probability theory plus noise model (Costello & Watts, 2014, 2016a, 2019Costello, Watts, & Fisher, 2018), making equivalent average predictions for simple events, conjunctions, and disjunctions. The Bayesian sampler does, however, make distinct predictions for conditional probabilities, and we show in a new experiment that this model better captures these judgments both qualitatively and quantitatively.…”

Probability Theory Plus Noise: Descriptive Estimation and Inferential Judgment

“…Our model thus predicts that this expression will have a value of 0, on average, in people's probability judgments just as required by standard probability theory. Exactly this pattern of agreement is seen in experimental results (Costello & Mathison, ; Costello & Watts, , , ; Fisher & Wolfe, ).…”