The statistics of cognitive variability: Explaining common patterns in individuals, groups and financial markets.

Psychological variability (i.e., "noise") displays interesting structure which is hidden by the common practice of averaging over trials. Interesting noise structure, termed 'stylized facts', is observed in financial markets (i.e., behaviors from many thousands of traders). Here we investigate the parallels between psychological and financial time series. In a series of three experiments (total N = 202), we successively simplified a market-based price prediction task by first removing external information, and then removing any interaction between participants. Finally, we removed any resemblance to an asset market by asking individual participants to simply reproduce temporal intervals. All three experiments reproduced the main stylized facts found in financial markets, and the robustness of the results suggests that a common cognitive-level mechanism can produce them. We identify one potential model based on mental sampling algorithms, showing how this general-purpose model might account for behavior across these very different tasks.

The autocorrelated Bayesian sampler: A rational process for probability judgments, estimates, confidence intervals, choices, confidence judgments, and response times.

Normative models of decision-making that optimally transform noisy (sensory) information into categorical decisions qualitatively mismatch human behavior. Indeed, leading computational models have only achieved high empirical corroboration by adding task-specific assumptions that deviate from normative principles. In response, we offer a Bayesian approach that implicitly produces a posterior distribution of possible answers (hypotheses) in response to sensory information. But we assume that the brain has no direct access to this posterior, but can only sample hypotheses according to their posterior probabilities. Accordingly, we argue that the primary problem of normative concern in decision-making is integrating stochastic hypotheses, rather than stochastic sensory information, to make categorical decisions. This implies that human response variability arises mainly from posterior sampling rather than sensory noise. Because human hypothesis generation is serially correlated, hypothesis samples will be autocorrelated. Guided by this new problem formulation, we develop a new process, the Autocorrelated Bayesian Sampler (ABS), which grounds autocorrelated hypothesis generation in a sophisticated sampling algorithm. The ABS provides a single mechanism that qualitatively explains many empirical effects of probability judgments, estimates, confidence intervals, choice, confidence judgments, response times, and their relationships. Our analysis demonstrates the unifying power of a perspective shift in the exploration of normative models. It also exemplifies the proposal that the "Bayesian brain" operates using samples not probabilities, and that variability in human behavior may primarily reflect computational rather than sensory noise. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

A unified explanation of variability and bias in human probability judgments: How computational noise explains the mean-variance signature.

Human probability judgments are both variable and subject to systematic biases. Most probability judgment models treat variability and bias separately: a deterministic model explains the origin of bias, to which a noise process is added to generate variability. But these accounts do not explain the characteristic inverse U-shaped signature linking mean and variance in probability judgments. By contrast, models based on sampling generate the mean and variance of judgments in a unified way: the variability in the response is an inevitable consequence of basing probability judgments on a small sample of remembered or simulated instances of events. We consider two recent sampling models, in which biases are explained either by the sample accumulation being further corrupted by retrieval noise (the Probability Theory + Noise account) or as a Bayesian adjustment to the uncertainty implicit in small samples (the Bayesian sampler). While the mean predictions of these accounts closely mimic one another, they differ regarding the predicted relationship between mean and variance. We show that these models can be distinguished by a novel linear regression method that analyses this crucial mean-variance signature. First, the efficacy of the method is established using model recovery, demonstrating that it more accurately recovers parameters than complex approaches. Second, the method is applied to the mean and variance of both existing and new probability judgment data, confirming that judgments are based on a small number of samples that are adjusted by a prior, as predicted by the Bayesian sampler. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Understanding the structure of cognitive noise.

Human cognition is fundamentally noisy. While routinely regarded as a nuisance in experimental investigation, the few studies investigating properties of cognitive noise have found surprising structure. A first line of research has shown that inter-response-time distributions are heavy-tailed. That is, response times between subsequent trials usually change only a small amount, but with occasional large changes. A second, separate, line of research has found that participants' estimates and response times both exhibit long-range autocorrelations (i.e., 1/f noise). Thus, each judgment and response time not only depends on its immediate predecessor but also on many previous responses. These two lines of research use different tasks and have distinct theoretical explanations: models that account for heavy-tailed response times do not predict 1/f autocorrelations and vice versa. Here, we find that 1/f noise and heavy-tailed response distributions co-occur in both types of tasks. We also show that a statistical sampling algorithm, developed to deal with patchy environments, generates both heavy-tailed distributions and 1/f noise, suggesting that cognitive noise may be a functional adaptation to dealing with a complex world.

Perceptual and Cognitive Judgments Show Both Anchoring and Repulsion.

One of the most robust effects in cognitive psychology is anchoring, in which judgments show a bias toward previously viewed values. However, in what is essentially the same task as used in anchoring research, a perceptual illusion demonstrates the opposite effect of repulsion. Here, we united these two literatures, testing in two experiments with adults (total N = 200) whether prior comparative decisions bias cognitive and perceptual judgments in opposing directions or whether anchoring and repulsion are two domain-general biases whose co-occurrence has so far gone undetected. We found that in both perceptual and cognitive tasks, anchoring and repulsion co-occur. Further, the direction of the bias depends on the comparison value: Distant values attract judgments, whereas nearby values repulse judgments. Because none of the leading theories for either effect account for both biases, theoretical integration is needed. As a starting point, we describe one such joint theory based on sampling models of cognition.

Clarifying the relationship between coherence and accuracy in probability judgments.

Bayesian approaches presuppose that following the coherence conditions of probability theory makes probabilistic judgments more accurate. But other influential theories claim accurate judgments (with high "ecological rationality") do not need to be coherent. Empirical results support these latter theories, threatening Bayesian models of intelligence; and suggesting, moreover, that "heuristics and biases" research, which focuses on violations of coherence, is largely irrelevant. We carry out a higher-power experiment involving poker probability judgments (and a formally analogous urn task), with groups of poker novices, occasional poker players, and poker experts, finding a positive relationship between coherence and accuracy both between groups and across individuals. Both the positive relationship in our data, and past null results, are captured by a sample-based Bayesian approximation model, where a person's accuracy and coherence both increase with the number of samples drawn. Thus, we reconcile the theoretical link between accuracy and coherence with apparently negative empirical results.

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

Human probability judgments are systematically biased, in apparent tension with Bayesian models of cognition. But perhaps the brain does not represent probabilities explicitly, but approximates probabilistic calculations through a process of sampling, as used in computational probabilistic models in statistics. Naïve probability estimates can be obtained by calculating the relative frequency of an event within a sample, but these estimates tend to be extreme when the sample size is small. We propose instead that people use a generic prior to improve the accuracy of their probability estimates based on samples, and we call this model the Bayesian sampler. 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, 2017; Costello & Watts, 2019; Costello, 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 distributions of probability estimates. We show in 2 new experiments that this model better captures these mean judgments both qualitatively and quantitatively; which model best fits individual distributions of responses depends on the assumed size of the cognitive sample. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Costly curiosity: People pay a price to resolve an uncertain gamble early.

Humans are inherently curious creatures, continuously seeking out information about future outcomes. Such advance information is often valuable, potentially allowing people to select better courses of action. In non-human animals, this drive for information can be so strong that they forego food or water to find out a few seconds earlier whether an uncertain option will provide a reward. Here, we assess whether people will exhibit a similar sub-optimal preference for advance information. Participants played a card-flipping task where they were probabilistically rewarded based on the pattern of 3 cards that were revealed after a 5-s delay. During this delay, participants could instead pay a cost to find out the next card's identity immediately. This choice to find out early did not influence the eventual outcome. Participants preferred to find out early about 80% of the time when the information was free; they were even willing to incur an expense to get advance information about the eventual outcome. The expected magnitude of the outcome, however, had little impact on the likelihood of finding out early. These results suggest that humans, like animals, value non-instrumental information and will pay a price for such information, independent of its utility.