Allowing robots to communicate naturally with humans is an important goal for social robotics. Most approaches have focused on building high-level probabilistic cognitive models. However, research in cognitive science shows that people often build common ground for communication with each other by seeking and providing evidence of understanding through behaviors like mimicry. Predictive State Representations (PSRs) allow one to build explicit, low-level models of the expected outcomes of actions, and are therefore well-suited for tasks that require providing such evidence of understanding. Using human-robot shadow puppetry as a prototype interaction study, we show that PSRs can be used successfully to both model human interactions, and to allow a robot to learn on-line how to engage a human in an interesting interaction.