The paper deals with state estimation of nonlinear stochastic dynamic systems. The state is estimated within the Bayesian framework using the Gaussian filter and the Gaussian mixture filter. The paper is concerned with the joint Gaussianity assumption of the Gaussian filter and monitoring its validity. For cases, in which the assumption becomes invalid, the paper proposes a structure adaptation of the filter by directional splitting of the Gaussian distribution to a Gaussian mixture distribution. Both the monitoring and the directional splitting are based on a non-Gaussianity measure. The proposed directional splitting is illustrated using a numerical example.