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Learning to perform a perceptual decision task is generally achieved through sessions of effortful practice with feedback. Here, we investigated how passive exposure to task-relevant stimuli, which is relatively effortless and does not require feedback, influences active learning. First, we trained mice in a sound-categorization task with various schedules combining passive exposure and active training. Mice that received passive exposure exhibited faster learning, regardless of whether this exposure occurred entirely before active training or was interleaved between active sessions. We next trained neural-network models with different architectures and learning rules to perform the task. Networks that use the statistical properties of stimuli to enhance separability of the data via unsupervised learning during passive exposure provided the best account of the behavioral observations. We further found that, during interleaved schedules, there is an increased alignment between weight updates from passive exposure and active training, such that a few interleaved sessions can be as effective as schedules with long periods of passive exposure before active training, consistent with our behavioral observations. These results provide key insights for the design of efficient training schedules that combine active learning and passive exposure in both natural and artificial systems. 

Abstract The significance and influence of U.S. Supreme Court majority opinions derive in large part from opinions’ roles as precedents for future opinions. A growing body of literature seeks to understand what drives the use of opinions as precedents through the study of Supreme Court case citation patterns. We raise two limitations of existing work on Supreme Court citations. First, dyadic citations are typically aggregated to the case level before they are analyzed. Second, citations are treated as if they arise independently. We present a methodology for studying citations between Supreme Court opinions at the dyadic level, as a network, that overcomes these limitations. This methodology—the citation exponential random graph model, for which we provide user-friendly software—enables researchers to account for the effects of case characteristics and complex forms of network dependence in citation formation. We then analyze a network that includes all Supreme Court cases decided between 1950 and 2015. We find evidence for dependence processes, including reciprocity, transitivity, and popularity. The dependence effects are as substantively and statistically significant as the effects of exogenous covariates, indicating that models of Supreme Court citations should incorporate both the effects of case characteristics and the structure of past citations.