Bayesian confidence for drift diffusion observers in dynamic stimuli tasks

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