Superorganisms such as social insect colonies are very successful relative to their non-social counterparts. Powerful emergent information processing capabilities would seem to contribute to the abundance of such ‘swarm’ systems, as they effectively explore and exploit their environment collectively. We develop a Bayesian model of collective information processing in a decision-making task: choosing a nest site (a ‘multi-armed bandit’ problem). House-hunting Temnothorax ants are adept at discovering and choosing the best available nest site for their colony: we propose that this is possible via rapid, decentralized estimation of the probability that each choice is best. Viewed this way, their behavioral algorithm can be understood as a statistical method that anticipates recent advances in mathematics. Our nest finding model incorporates insights from approximate Bayesian computation as a model of colony-level behavior; and particle filtering as a model of Temnothorax ‘tandem running’. Our framework suggests that the mechanisms of complex collective behavior can sometimes be explained as a spatial enactment of Bayesian inference. It facilitates the generation of quantitative hypotheses regarding individual and collective movement behaviors when collective decisions must be made. It also points to the potential for bioinspired statistical techniques. Finally, it suggests simple mechanisms for collective decision-making in engineered systems, such as robot swarms.