Figure 2. Bayesian observer models for the perceptual task sequence.

(a) The discrimination judgment does not affect the estimated stimulus orientation for an observer who considers both tasks independently. (b) In contrast, the self-consistent observer imposes a causal dependency such that the judgment in the discrimination task (e.g., 'cw’) conditions the estimation process in form of a choice-dependent prior. It effectively sets the posterior probability to zero for any orientation value that is inconsistent with the preceding discrimination judgment. The truncated posterior distribution, together with a loss function that penalizes larger estimation errors stronger than smaller ones, leads to the characteristic bimodal distribution pattern. Note, however, that this basic formulation is not quite sufficient to explain some details of the estimation data (Figure 1d).