Inferring the causes of noise from binary outcomes: A normative theory of learning under uncertainty

Inferring the true cause of noise—distinguishing between volatility (environmental change) andstochasticity (outcome randomness)—is essential for learning in noisy environments. Whilemost studies rely on binary outcomes, previous models are designed for continuous outcomeand use ad hoc approximations to handle binary data, introducing theoretical inconsistenciesand interpretational issues. Here, we develop a normative framework for inferring the causes ofnoise from binary feedback that remains faithful to the discrete nature of the generative processand underlying statistical structure. First, we establish a generative model using a state spaceapproach tailored for binary outcomes and derive the corresponding hidden Markov modelinference procedure. Second, we introduce a computational model combining the hiddenMarkov model with particle filtering to simultaneously infer volatility and stochasticity frombinary outcomes. Third, we validate predictions through a 2×2 probabilistic reversal learningtask with human participants, systematically manipulating both noise parameters. Results showthat participants adjust their learning rates consistent with model predictions, increasinglearning rates under volatile conditions and decreasing them under high stochasticity. Ourtheoretical and experimental results offer a principled approach for dissociating volatility andstochasticity from binary outcomes, providing insights into learning processes relevant totypical cognition and psychiatric conditions.