This paper presents a probabilistic model-based approach to centralized multi-agent trajectory planning. This approach allows for incorporating uncertainty of the state and dynamics of the agents directly in the model. Probabilistic inference is then efficiently automated using message passing. The recently introduced normal-with-unknown-variance (NUV) priors are used to prevent collisions between agents and obstacles. Furthermore, a new expectation-maximization inference scheme is derived for box and half-space NUV priors, which takes state uncertainty into account when avoiding collisions.