Studies of infant looking times over the past 50 years have provided profound insights about cognitive development, but their dependent measures and analytic techniques are quite limited. In the context of infants' attention to discrete sequential events, we show how a Bayesian data analysis approach can be combined with a rational cognitive model to create a rich data analysis framework for infant looking times. We formalize (i) a statistical learning model, (ii) a parametric linking between the learning model's beliefs and infants' looking behavior, and (iii) a data analysis approach and model that infers parameters of the cognitive model and linking function for groups and individuals. Using this approach, we show that recent findings from Kidd, Piantadosi and Aslin (2012) of a U-shaped relationship between look-away probability and stimulus complexity even holds within infants and is "not" due to averaging subjects with different types of behavior. Our results indicate that individual infants prefer stimuli of intermediate complexity, reserving attention for events that are moderately predictable given their probabilistic expectations about the world.