Smooth pursuit eye movements respond on the basis of both immediate and anticipated target motion, where anticipations may be derived from either memory or perceptual cues. To study the combined influence of both immediate sensory motion and anticipation, subjects pursued clear or noisy random dot kinematograms (RDKs) whose mean directions were chosen from Gaussian distributions with SDs = 10° (narrow prior) or 45° (wide prior). Pursuit directions were consistent with Bayesian theory in that transitions over time from dependence on the prior to near total dependence on immediate sensory motion (likelihood) took longer with the noisier RDKs and with the narrower, more reliable, prior. Results were fit to Bayesian models in which parameters representing the variability of the likelihood either were or were not constrained to be the same for both priors. The unconstrained model provided a statistically better fit, with the influence of the prior in the constrained model smaller than predicted from strict reliability-based weighting of prior and likelihood. Factors that may have contributed to this outcome include prior variability different from nominal values, low-level sensorimotor learning with the narrow prior, or departures of pursuit from strict adherence to reliability-based weighting. Although modifications of, or alternatives to, the normative Bayesian model will be required, these results, along with previous studies, suggest that Bayesian approaches are a promising framework to understand how pursuit combines immediate sensory motion, past history, and informative perceptual cues to accurately track the target motion that is most likely to occur in the immediate future.NEW & NOTEWORTHY Smooth pursuit eye movements respond on the basis of anticipated, as well as immediate, target motions. Bayesian models using reliability-based weighting of previous (prior) and immediate target motions (likelihood) accounted for many, but not all, aspects of pursuit of clear and noisy random dot kinematograms with different levels of predictability. Bayesian approaches may solve the long-standing problem of how pursuit combines immediate sensory motion and anticipation of future motion to configure an effective response.