*2.4. Gaussian Mixture Models*

The mixture model is a probability model that can be used to represent an overall distribution with K sub-distributions. In other words, the mixture model represents the probability distribution of observational data overall, which is a mixture of K subdistributions. The mixture model does not require the observational data to provide information about the sub-distributions to calculate the probability that the observational data are in the overall distribution.

In general, a mixture model can use any probability distribution, but due to the good mathematical properties and good computational performance of the Gaussian distribution, the Gaussian mixture model is the most widely used model in practice [17].

**Definition 7.** *The probability distribution of Gaussian mixture models is*

$$P(\mathbf{x} \mid \Theta) = \sum\_{i=1}^{K} \alpha\_i p\_i(\mathbf{x} \mid \theta\_i), \tag{7}$$

*where* <sup>Θ</sup> <sup>=</sup> (*α*1,..., *<sup>α</sup>K*, *<sup>θ</sup>*1,..., *<sup>θ</sup>K*) *such that <sup>α</sup><sup>i</sup>* <sup>≥</sup> 0, <sup>∑</sup>*<sup>K</sup> <sup>i</sup>*=<sup>1</sup> *α<sup>i</sup>* = 1*, α<sup>i</sup> is the probability that the observational data belong to the i-th submodel and pi is the Gaussian distribution density function of the i-th submodel, whose parameter is θi.*
