The probabilistic interpretation of the dynamics of the sources in the standard NSSM is

where is the nonlinear dynamical mapping and is the covariance matrix of the zero mean innovation process . The typical linear approach, applied to the nonlinear case, would use a different and for all the different states of the HMM.

The simplified model uses only one dynamical mapping but has an own covariance matrix for each HMM state . In addition to this, the innovation process is not assumed to be zero-mean but it has a mean depending on the HMM state. Mathematically this means that

where is the HMM state, and are, respectively, the mean and the covariance matrix of the innovation process for that state. The prior model of remains unchanged.

Equation (5.50) summarises the differences between the switching NSSM and its components, as they were presented in Sections 5.1 and 5.2. The HMM ``output'' distribution is the one defined in Equation (5.50), not the data likelihood as in the ``stand-alone'' model. Similarly the model of the continuous hidden states in NSSM is slightly different from the one specified in Equation (5.25).