Fig. 6: POMDP model explains different values for \({d}^{\prime}\) and meta-\({d}^{\prime}\), and different patterns of confidence in reaction-time experiments.

a A signal detection theory framework predicts identical \({d}^{\prime}\) and meta-\({d}^{\prime}\) when the same observations inform both choice and confidence rating. The competing stimuli (e.g., right and leftward motion of a particular coherence) give rise to two observation distributions. The c₁ criterion is used for choosing right and left, and the c₂ criteria are used to report low or high confidence for each choice. \({d}^{\prime}\) quantifies type 1 sensitivity: the distance between the distributions in units of standard deviation. Meta-\({d}^{\prime}\) quantifies type 2 sensitivity: the separation of the two distributions compatible with hit rate and false alarm rate of confidence reports: p (high conf∣correct) and \(p\left(\,{{\mbox{high conf}}}| {{\mbox{incorrect}}}\,\right)\), respectively. In SDT, \({d}^{\prime}\) and c₁ fully constrain meta-\({d}^{\prime}\) and an optimal meta-cognitive observer must have equal \({d}^{\prime}\) and meta-\({d}^{\prime}\)³⁰. b Observation noise could cause highly variable μ_t at the beginning of a trial, and thus temporarily produce excessive confidence. This excessive confidence may become permanent if the decision-making process is stopped by reaching the termination bounds. Solid white lines show the two decision termination bounds (observation cost, 10⁻⁴). Thresholds for separating low and high confidence ratings are shown as boundaries between blue (low confidence) and red (high confidence) regions. The horizontal dashed line shows the boundary that separates right and left direction choices based on the sign of the inferred coherence. Yellow dots and lines show mean ± 2 × s. d. (95% of the distribution mass) of the inferred coherence for a particular stimulus strength (c = +12.8%) at a few different time steps (10 ms per step). Temporary excessive confidence due to early termination is more prominent for the incorrect trials (negative μ_t in this simulation). c Early termination can cause a modest reduction of accuracy and a marked increase of high-confidence ratings. d Early termination can cause a larger increase in the probability of high confidence ratings for incorrect than correct choices. e Changes in accuracy (c) and confidence ratings (d) can lead to a larger drop in meta-\({d}^{\prime}\) than \({d}^{\prime}\). Model parameters are identical to those for monkey M1, except for the observation cost. f For experiments with simultaneous reports of choice and confidence, our model predicts higher confidence for incorrect choices on trials with stronger stimuli (red dashed line). This pattern is partly caused by lower decision times for stronger stimuli and the dependence of model confidence on elapsed time (Fig. 3c). g In contrast, for sequential reports of choice and confidence, our model predicts reduced confidence for incorrect choices for stronger stimuli (red dashed line). This is due to sensory and motor delays that render the last observations inconsequential for the choice but the model uses those observations to refine its confidence report following the choice.