The Belief Update Gate: Separating Inertia from Learning in Human-AI Interaction
Repeated human-AI interaction is often analyzed through pooled belief-updating slopes: users observe AI successes and failures, revise reported beliefs in the feedback-consistent direction, but appear conservative on average. We show that such averages can obscure an important distinction between whether an elicited belief report changes at all and how it changes conditional on movement. We refer to this measurement-aware decomposition as the belief update gate. Reanalyzing a multi-task human-AI decision-making dataset with 240 participants, 7,200 trials, and three task domains, we find substantial non-movement in reported beliefs: 67.3% of trial-level belief changes are exactly zero, and 76.4% are smaller than five percentage points. Separating non-moving from moving reports changes the descriptive interpretation of pooled conservatism: the within-trajectory slope rises from 0.494 overall to 0.949 among rows with nonzero movement. Since this latter estimate conditions on observed movement, we interpret it as a descriptive decomposition rather than as evidence of a near-Bayesian latent learning process. Complementary hurdle style analyses (i.e., modeling zero vs. non-zero changes before predicting update magnitude) show that the absolute discrepancy between feedback and entering belief predicts whether a report changes, while the signed feedback discrepancy predicts the direction and magnitude of change among reports that move. Importantly, observed non-movement does not distinguish genuine latent belief inertia from small unexpressed updates, rounding, or other reporting processes. These findings show that calibration analyses of repeated human--AI interaction should distinguish visible non-movement in elicited belief reports from updating conditional on movement rather than treating reported beliefs as a single continuous updating process.