- The problem of sequential Bayesian estimation in linear non-Gaussian problems is addressed. In the Gaussian sum filter (GSF), the non-Gaussian system noise, the measurement noise, and the posterior state densities are modeled by the Gaussian mixture model (GMM). The GSF is optimal under the minimum-mean-square error (MMSE) criterion, however it is impractical due to the exponential model order growth of the system probability density function (pdf). The proposed recursive estimator, named the Gaussian mixture Kalman filter (GMKF), combines the GSF and the model order reduction procedure. The posterior state density at each iteration is approximated by a lower order density. This model order reduction procedure minimizes the estimated Kullback-Leibler divergence (KLD) of the reduced order density from the original density at each step. The estimation performance of the proposed GMKF is compared with the interactive multiple modeling (IMM), particle filter (PF), Gaussian sum PF (GSPF), and the GSF with mixture reduction (MR) method via simulations. It is shown in several examples that the proposed GMKF outperforms the other tested algorithms in terms of estimation accuracy. The superior estimation performance of the GMKF is obtained at the expense of its computational complexity, which is higher than the IMM and the MR algorithms.