*2.1. Review of MCLUST Model*

**Finite mixture model** Let *f*1(*y*;Θ1), *f*2(*y*;Θ2), ..., *fG*(*y*;Θ*G*) be *G* probability distributions defined on the *d*-dimensional random vector *y*, and a mixture of the *G* distributions is formed by taking proportions {*τk*} of the population from components { *fk*}, with probability density given by

$$f(\mathbf{y}; \boldsymbol{\Theta}) = \sum\_{k=1}^{G} \tau\_k f\_k(\mathbf{y}; \boldsymbol{\Theta}\_k)\_\prime \tag{1}$$

where Θ = (Θ1, ...,Θ*G*) are model parameters.

**Component density** The MCLUST model assumes that the distribution of each *y* is a mixture of multivariate normal distributions. Under the MCLUST model, the component density of *y* in group *k* is 

$$f\_k(\mathbf{y}; \boldsymbol{\mu}\_k, \boldsymbol{\Sigma}\_k) = \frac{\exp\left\{-\frac{1}{2} (\boldsymbol{\Psi} - \boldsymbol{\mu}\_k)^T \boldsymbol{\Sigma}\_k^{-1} (\boldsymbol{\Psi} - \boldsymbol{\mu}\_k)\right\}}{\sqrt{\det[2\pi \boldsymbol{\Sigma}\_k]}},\tag{2}$$

In other words,

$$\|\mathbf{y}\|k \sim \mathcal{N}\_d(\mathfrak{\mu}\_k, \mathbf{E}\_k). \tag{3}$$

The (marginal) probability density of *y* is given by

$$f(\mathbf{y}) = \sum\_{k=1}^{G} \tau\_k f\_k(\mathbf{y}; \boldsymbol{\mu}\_k, \boldsymbol{\Sigma}\_k). \tag{4}$$

**Likelihood function** Suppose a sample of *n* independent and identically distributed (iid) random vectors *y* = (*y*1, ..., *yn*) is drawn from the mixture. The (observed) log likelihood of the sample is then

$$d\_O(\Theta; \mathfrak{y}) = \sum\_{i=1}^n \log f(\mathfrak{y}\_i) = \sum\_{i=1}^n \log \sum\_{k=1}^G \tau\_k f\_k(\mathfrak{y}\_i; \mathfrak{h}\_k, \mathfrak{L}\_k), \tag{5}$$

where Θ = (*τ*1, ..., *τG*;*μ*1, ...,*μG*;Σ1, ...,Σ*G*) are the model parameters.
