The Predictive Linear Gaussian model (or PLG) improves upon traditional linear dynamical system models by using a predictive representation of state, which makes consistent parameter estimation possible without any loss of modeling power and while using fewer parameters. This work extends the PLG to model nonlinear dynamical systems through the use of a kernelized, nonlinear mixture technique. The resulting generative model has been named the "MPLG," for "Mixture of PLGs." We also develop a novel technique to perform inference in the model, which consists of a hybrid of sigma-point approximations and analytical statistics. We show that the technique leads to fast and accurate approximations, and that it is general enough to be applied in other contexts. We empirically explore the MPLG and demonstrate its viability on several real-world and synthetic tasks.