The Autocorrelated Bayesian Sampler: A Rational Process for Probability Judgments, Estimates, Confidence Intervals, Choices, Confidence Judgments, and Response Times

Abstract

Estimation, choice, confidence, and response times are the primary behavioural measures in perceptual and cognitive tasks. These measures have attracted extensive modeling efforts in the cognitive sciences, but there is the lack of a unified approach to explain all measures simultaneously within one framework. We propose an Autocorrelated Bayesian Sampler (ABS), assuming that people sequentially sample from a posterior probability distribution of hypotheses on each trial of a perceptual or cognitive task. Unlike most accounts of choice, we do not assume that the system has global knowledge of the posterior distribution. Instead it uses a sophisticated sampling algorithm to make do with local knowledge, and so produces autocorrelated samples. The model assumes that each sample takes time to generate, and that samples are used by well-validated mechanisms to produce estimates, choices, and confidence judgments. This relatively simple framework clears a number of well-known empirical hurdles for models of choice, confidence, and response time. The autocorrelation be- tween samples also allows the model to predict the long-range between-trial dependence observed in both estimates and response times.

Noise in Cognition: Bug or Feature?

“…The multiple chains of MC 3 introduce human-like long-range autocorrelations (see Figure 3D 3 ), and surprisingly, in both fixed target and random walk target tasks, this local sampling algorithm also produces almost no autocorrelations in the changes between estimates, but does show autocorrelations in the magnitude of these changes (see Figure 3E 3 ; Zhu, Sundh, et al, 2021). MC 3 better fit the overwhelming majority of participants in a price prediction task than non-sampling models of human behavior (Spicer, Zhu, et al, 2022a).…”

“…This viewpoint predicts that the brain will roughly follow Bayesian probability theory, but will be subject to systematic biases due to computational constraints on cognition limiting the number of samples drawn. These biases will arise in a variety of ways: first, small samples will depend on their starting point (because the choice of starting point will only 'wash out' after a large sequence of samples has been drawn, so that the entire probability space has been explored) -this dependence on starting point has been argued to account for anchoring and adjustment, sub-and super-additivity, among other effects (Dasgupta et al, 2017;Lieder et al, 2018;Sanborn & Chater, 2016;Spicer, Zhu, et al, 2022b;Zhu, Sundh, et al, 2021). Moreover, the brain needs to make appropriate inferences in the light of small samples (for example, not simply assuming that an event which happens, say, twice in a sample of two must have a probability of one).…”

“…Appropriate correction for small samples leads to regression of probability values from extreme values, which can provide an explanation for the conjunction fallacy, among other effects (Zhu et al, 2020;Zhu, Sundh, et al, 2021). Moreover, given small samples, the brain has to extrapolate the probability of new items based on their similarity to the small number of items that have been sampled (in computational statistics, a method known as Approximate Bayesian Computation), which provides an elegant explanation of the close relationship between judgments of probability and similarity that underlies the representativeness heuristic (Kahneman & Tversky, 1972).…”

“…Randomly selecting a rule on each trial is a form of sampling too. Additionally, random walk models can be formulated from sampling hypotheses instead of data, providing a link between these approaches and drift diffusion models (Hamrick et al, 2015;Vul et al, 2014;Zhu, Sundh, et al, 2021). Indeed, regarding the observations about cognitive noise we reviewed above, Findling and Wyart (2021) commented that sampling was a potential contributor to the computational noise found in their task.…”

“…The remaining chains are useful for exploring the distribution, because, when appropriate, one of these remaining chains can swap positions with the MCMC chain sampling from the correct target distribution. In this way, MC 3 can make occasional large jumps across the hypothesis space, allowing it also produce heavy-tailed changes even when sampling from a fixed Gaussian distribution (see Figure 3C 3 ; Zhu et al, 2022), or when the target follows a Gaussian random walk in either time interval estimation tasks or price prediction tasks (Spicer, Zhu, et al, 2022a;Zhu, Sundh, et al, 2021).…”