Abstract Gaia DR2 provides an unprecedented sample of stars with full 6D phase-space measurements, creating the need for a self-consistent means of discovering and characterising the phase-space overdensities known as moving groups or associations. Here we present Chronostar, a new Bayesian analysis tool that meets this need. Chronostar uses the Expectation-Maximisation algorithm to remove the circular dependency between association membership lists and fits to their phase-space distributions, making it possible to discover unknown associations within a kinematic data set. It uses forward-modelling of orbits through the Galactic potential to overcome the problem of tracing backward stars whose kinematics have significant observational errors, thereby providing reliable ages. In tests using synthetic data sets with realistic measurement errors and complex initial distributions, Chronostar successfully recovers membership assignments and kinematic ages up to ≈100 Myr. In tests on real stellar kinematic data in the phase-space vicinity of the β Pictoris Moving Group, Chronostar successfully rediscovers the association without any human intervention, identifies 15 new likely members, corroborates 43 candidate members, and returns a kinematic age of 17.8 ± 1.2 Myr. In the process we also rediscover the Tucana-Horologium Moving Group, for which we obtain a kinematic age of $36.3^{+1.3}_{-1.4}$ Myr.