Gaussian and student-t filtering using implicit measurements via variational bayes

Kalman-type filters assume that the measurements are described explicitly as a function of the state. However, the state and measurement may be related implicitly by an equation that could not be solved for the measurement in closed form. We introduce recursive estimators for nonlinear discrete-time state models with implicit measurements in order to overcome such difficulties. Our estimators are based on the Gaussian Filtering model, extending well known nonlinear Kalman-type filters. We further define outlier-robust filters by modeling the implicit measurement equation with a multivariate Student-t distribution. We approximate the posterior distributions of the filter equations via Variational Bayes. Preliminary results with a simulated model validate our filters.