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We propose a new approach to approximating the Chapman-Kolmogorov equation (CKE) for particle-based nonlinear filtering algorithms, using a new proposal distribution and the improved fast Gauss transform (IFGT) with tighter accuracy bounds. The new proposal distribution, used to obtain a Monte Carlo (MC) approximation of the CKE, is based on the proposal distribution found in the auxiliary marginal particle filter (AMPF). By using MC integration to approximate the integrals of the AMPF proposal distribution as well as the CKE, we demonstrate significant improvement in terms of both error and computation time. We consider the additive state noise case where the evaluation of the CKE is equivalent to performing kernel density estimation (KDE), thus fast methods such as the IFGT can be used. In practice, the IFGT demonstrates performance far better than that which is predicted by current error analysis, therefore the existing bounds are not useful for determining the IFGT parameters which in practice have to be chosen experimentally in order to obtain satisfactory compromise between accuracy and speed of the filtering algorithm. We provide in this paper much improved performance bounds for the IFGT, and which unlike the previous bound, are consistent with the expectation that the error decreases as the truncation order of the IFGT increases. The new bounds lead to a new definition of the IFGT parameters and give important insight into the effect that these parameters have on the error, therefore facilitating the choice of parameters in practice. The experimental results show that we can obtain similar error to the sequential importance sampling (SIS) particle filter, while using fewer particles. Furthermore the choice of the new IFGT parameters remains roughly the same for all the examples that we give.