**4. Parameter Estimation**

From the previous section, it is now clear how the hidden state transfers and how to compute the observation probability. In this section, we focus on how to obtain the filtering and smoothing probabilities, and to apply parameter updating based on the on-line EM algorithm.

Before starting the explanation, we need to introduce the parameter set first. As described in the previous Section, the parameter set can be defined as *θ* = {*ζk*, *alk*, *cij*, *μij*, Σ*ij*}, in which *ζ<sup>k</sup>* is the initial probability of hidden state, and *alk* is the *l*-th row and *k*-th column element in the transition matrix *<sup>A</sup>*. Because GMM density only depends on *<sup>v</sup>n*, then *<sup>i</sup>* <sup>∈</sup> <sup>Λ</sup> <sup>×</sup> <sup>Γ</sup>, *<sup>j</sup>* <sup>∈</sup> *<sup>K</sup>*. In SemiTMC-GMM, the entire hidden state is (*V*, *<sup>D</sup>*), and then *<sup>l</sup>*, *<sup>k</sup>* <sup>∈</sup> <sup>Λ</sup> <sup>×</sup> <sup>Γ</sup> <sup>×</sup> *<sup>L</sup>*, and *<sup>l</sup>*, *<sup>k</sup>* is equal to the couple of (*i*, *dn*). Therefore, the initial probability becomes *<sup>ζ</sup><sup>k</sup>* <sup>=</sup> *<sup>p</sup>*((*v*1, *<sup>d</sup>*1) = *<sup>k</sup>*), and *alk* <sup>=</sup> *<sup>p</sup>*((*vn*<sup>+</sup>1, *dn*+1) = *<sup>k</sup>*|(*vn*, *dn*) = *<sup>l</sup>*). For simplification, the indices *i*, *j*, *l*, *k* will keep the same meaning and will no longer be specified in the remaining.
