*3.2. TMC Embedding a Gaussian Mixture Model*

When extending TMC to TMC-GMM, it needs to introduce Gaussian mixture density into the conditioned observation probability. In fact, embedding GMM in TMC can be regarded as introducing a new statistic process *<sup>H</sup>* = (*H*1, ··· , *HN*) into TMC, where *Hn* takes its value *hn* in a finite set *<sup>K</sup>* <sup>=</sup> {1, ··· , *<sup>κ</sup>*} and *<sup>κ</sup>* is the number of Gaussian components in the mixture. Please remind that *<sup>H</sup>* has no realistic meaning, it is just a latent variable in the model to introduce the mixtures. Let *cij* be the weight of *j*th Gaussian mixture component when *v<sup>n</sup>* = *i*, with the constraint ∑*<sup>κ</sup> <sup>j</sup>*=<sup>1</sup> *cij* = 1. *μij* and Σ*ij* are the mean value and covariance of the Gaussian mixture component. Denoting *Z* = (*T*, *H*) and assuming *p*(*h<sup>N</sup>* <sup>1</sup> <sup>|</sup>*v<sup>N</sup>* <sup>1</sup> ) = *<sup>N</sup>* ∏ *n*=1 *<sup>p</sup>*(*hn*|*vn*), *<sup>Z</sup>* is Markovian with transitions:

$$p(\boldsymbol{z}\_{n+1}|\boldsymbol{z}\_n) = p(\boldsymbol{\upsilon}\_{n+1}|\boldsymbol{\upsilon}\_n)p(h\_{n+1}|\boldsymbol{\upsilon}\_{n+1})p(\boldsymbol{y}\_{n+1}|\boldsymbol{\upsilon}\_{n+1}, h\_{n+1}),\tag{6}$$

where

$$p(y\_n | \mathbf{v}\_n = i, h\_n = j) \sim \mathcal{N}\left(\mu\_{i\bar{j}}, \Sigma\_{i\bar{j}}\right), \quad i \in \Lambda \times \Gamma, j \in \mathcal{K},\tag{7a}$$

$$p(\boldsymbol{y}\_n|\boldsymbol{\sigma}\_n) = \sum\_{j=1}^{k} \boldsymbol{c}\_{ij} \cdot p(\boldsymbol{y}\_n|\boldsymbol{\sigma}\_n = i, h\_n = j),\tag{7b}$$

with *<sup>p</sup>*(*hn* <sup>=</sup> *<sup>j</sup>*|*v<sup>n</sup>* <sup>=</sup> *<sup>i</sup>*) = *cij*. We can see that Equations (6) and (7a) are extensions of Equations (2) and (3), by introducing a new process *H*. The dependency graph of TMC-GMM is shown in Figure 1b.

Please notice that the only difference between TMC and TMC-GMM is the Gaussian densities in TMC are replaced with Gaussian mixtures, all the other calculations remain the same. Then estimating the individual *xn* and *un* in TMC-GMM follows the same as in TMC, by using Equations (4) and (5).
