Expressions for Bayesian confidence of drift diffusion observers in dynamic stimuli tasks

Abstract

6Much work has explored the possibility that the drift diffusion model, a model of response times and choices, could 7 be extended to account for confidence reports. Many methods for making predictions from such models exist, 8 although these methods either assume that stimuli are static over the course of a trial, or are computationally 9 expensive, making it difficult to capitalise on trial-by-trial variability in dynamic stimuli. Using the framework of 10 the drift diffusion model with time-dependent thresholds, and the idea of a Bayesian confidence readout, we derive 11 expressions for the probability distribution over confidence reports. In line with current models of confidence, 12 the derivations allow for the accumulation of "pipeline" evidence which has been received but not processed 13 by the time of response, the effect of drift rate variability, and metacognitive noise. The expressions are valid 14 for stimuli which change over the course of a trial with normally distributed fluctuations in the evidence they 15 provide. A number of approximations are made to arrive at the final expressions, and we test all approximations 16 via simulation. The derived expressions only contain a small number of standard functions, and only require 17evaluating once per trial, making trial-by-trial modelling of confidence data in dynamic stimuli tasks more feasible. 18 We conclude by using the expressions to gain insight into the confidence of optimal observers, and empirically 19 observed patterns.

Bayesian confidence in optimal decisions

“…Instead we fit the models to confidence, given the stimulus, response and response time. Fitting to confidence reports allows us to use simple mathematical expressions, which are computationally cheap to evaluate (Calder-Travis et al, 2020). Response times and responses effectively become held out data, which the models are not fit too.…”

“…We want to capture the idea that trial-to-trial quality of information processing varies, such that the effect of presented evidence on the accumulator varies. We use the term "standardised drift-rate" to refer to a coefficient which has a multiplicative effect on the strength of the relationship between evidence presented and changes in the accumulator (Appendix B;Calder-Travis et al, 2020). When standardised drift-rate is 1, the effect of evidence on the accumulator is at its average level.…”

“…These findings can be reconciled by considering the timing of decision and confidence reports (Calder-Travis et al, 2020;Desender, Donner & Verguts, 2020;Kiani et al, 2014…”

“…Faster responses are associated with reduced difficulty discounting, and greater confidence. Hence, in the absence of pipeline evidence, confidence in errors will be greater when a task is easier (Calder-Travis et al, 2020;Desender et al, 2020). When confidence reports follow a decision, and pipeline evidence is accumulated, pipeline evidence generally favours the correct option, reducing confidence in errors.…”

“…Next we derive qualitative predictions from the model variants, which we test using a previously collected data set, and in a new preregistered study. We then use newly developed, computationally cheap expressions to fit the models using trial-by-trial predictions for confidence (Calder-Travis, Bogacz & Yeung, 2020). This allows us to examine whether any model can account for the precise quantitative effects observed in confidence data.…”