Sequential inference methods have played a crucial role in many of the technological marvels that we use today, from GPS and navigation systems to machine learning. Most current methods, such as the unscented Kalman filter (UKF) make several, occasionally crippling assumptions which allow them to work efficiently and accurately for approximately linear dynamics. The problem with this is that the majority of systems are not linear. Inference methods fully representing the dynamics and probability distributions were considered infeasible in the early days of sequential inference. However, with the capabilities of modern computers this is no longer the case. In this thesis we propose a method to evolve a probability distribution on a dynamical system explicitly. This is done by using a finite volume partial differential equation solver to solve the continuity equation, combined with Bayesian observations. We present an example case of the simple pendulum and compare this with the UKF to examine several advantages.
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