This paper considers the filtering problem for discrete-time linear systems where the distributions of the process and observation noises are of gaussian sum distributions. Since the gaussian sum noise can be considered to be a sample from one of the gaussian distributions forming the gaussian sum, we define the distribution selection parameters that specify sample noises from the gaussian sum distribution. By using the maximuma posteriori(MAP) estimates of the selection parameters, a robust state estimation algorithm combined with the Kalman filter is developed. Simulation studies are also included to show the effectiveness of the present algorithm.