*2.2. MCLUST-ME Model*

We extend the MCLUST model by associating each data point with an error term and assumes that the covariance matrix of each error term is either known or can be estimated.

**Component density** Given that *y* belongs to component *k*, the MCLUST-ME models assumes that there exists a latent variable *w*, representing its "truth" part, and , representing its "error" part, such that

$$\begin{cases} \mathbf{y} = \mathbf{w} + \mathbf{e}, \\ \mathbf{w}|k \sim N\_d(\mu\_k, \mathbf{E}\_k), \\ \mathbf{e} \sim N\_d(\mathbf{0}, \mathbf{A}), \end{cases} \tag{6}$$

where *w* and are independent. *μ<sup>k</sup>* and Σ*<sup>k</sup>* are unknown mean and covariance parameters (same as in the MCLUST model), and Λ is the known error covariance matrix associated with *y*. The distribution of *y* being in component *k* is then

$$\mathbf{y}|k \sim \mathcal{N}\_d(\boldsymbol{\mu}\_k, \boldsymbol{\Sigma}\_k + \boldsymbol{\Lambda}),\tag{7}$$

with density function

$$\mathcal{G}\_{\mathcal{S}k}(\mathbf{y}; \boldsymbol{\mu}\_k, \boldsymbol{\Sigma}\_k, \mathbf{A}) = \frac{\exp\left\{-\frac{1}{2} (\boldsymbol{\mathfrak{y}} - \boldsymbol{\mu}\_k)^T \left(\boldsymbol{\Sigma}\_k + \boldsymbol{\mathsf{A}}\right)^{-1} \left(\boldsymbol{\mathfrak{y}} - \boldsymbol{\mu}\_k\right)\right\}}{\sqrt{\det[2\pi(\boldsymbol{\Sigma}\_k + \boldsymbol{\mathsf{A}})]}},\tag{8}$$

and the (marginal) probability density of *y* is given by

$$\chi\_{\mathcal{S}}(\mathbf{y}) = \sum\_{k=1}^{G} \tau\_{k} \mathbb{S}\_{k}(\mathbf{y}; \boldsymbol{\mu}\_{k}, \boldsymbol{\Sigma}\_{k}, \boldsymbol{\mathsf{A}}). \tag{9}$$

**Likelihood function** Suppose a sample of *n* iid random vectors *y* = (*y*1, ..., *yn*) is drawn from the mixture, where each *yi* is associated with known error covariance matrix Λ*i*. The (observed) log likelihood of the sample is then

$$d\_O(\boldsymbol{\Theta}; \mathbf{y}) = \sum\_{i=1}^{n} \log g(\mathbf{y}\_i) = \sum\_{i=1}^{n} \log \sum\_{k=1}^{G} \tau\_k g\_k(\mathbf{y}\_i; \boldsymbol{\mu}\_k, \boldsymbol{\Sigma}\_k, \boldsymbol{\Lambda}\_i), \tag{10}$$

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

In summary, the MCLUST-ME and MCLUST models have the same set of model parameters for the normal components and the mixing proportions. The key difference is that under the MCLUST-ME model, the measurement or observation errors of the observations are explicitly modeled, and observations are each associated with a given error covariance matrix.
