In this paper, we develop a novel method for approximate continuous-discrete Bayesian filtering. The projection filtering framework is exploited to develop accurate approximations of posterior distributions within parametric classes of probability distributions. This is done by formulating an ordinary differential equation for the posterior distribution that has the prior as initial value and hits the exact posterior after a unit of
time. Particular emphasis is put on exponential families, especially the Gaussian family of densities. Experimental results demonstrate the efficacy and flexibility of the method.