  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 ${\beta}$ 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.
