On fusion of sensor measurements and observation with uncertain timestamp for target tracking

We consider a target tracking problem where, in addition to the usual sensor measurements, accurate observations with uncertain timestamps are available. Such observations could, e.g., come from traces left by a target or from witnesses of an event, and have the potential in some scenarios to improve the accuracy of an estimate significantly. The Bayesian solution to the smoothing problem for one observation with uncertain timestamp is derived for a linear Gaussian state space model. The joint and marginal distributions of the states and uncertain time are derived, as well as the minimum mean squared error (MMSE) and maximum a posteriori (MAP) estimators. To attain an intuition for the problem in consideration a simple first-order example is presented and its posterior distributions and point estimators are compared and examined in some depth.