of the data set.

Mixture-of-Gaussians model for the field sources {#MG}
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We began by modelling the distribution of foreground and background sources in the representation space with a mixture-of-Gaussians model. The mixture-of-Gaussians model is a simple linear combination of $k_1$ multivariate normal (Gaussian) distributions, with the index 1 refering to the set of field contaminants (stars mostly, although it may contain one or various types of extragalactic sources, an issue that will be pursued further in future works).

A model is given by the $k_1$ means ($\vec{\mu}_{1,j}, j=1,2,...,k_1$) and covariance matrices ($\Sigma_{1,j}, j=1,2,...,k_1$) of the multivariate normal components in the mixture. We denote this set of parameters as $\vec\theta_1$. Below we use the subscript $j$ for indexing mixture components. Let $\vec{m}$ be the vector of observations (the subset of proper motions, apparent magnitudes and colour indices) used to represent sources. Then, the mixture-of-Gaussians model of the likelihood for the field sources is defined as

$$p(\vec{m}|{\cal C}_1)= \sum_{j=1}^{k_1} \pi_{1,j} \cdot {\cal
  N}_{\mathbf{\mu_{1,j}},\Sigma_{1,j}}(\vec{m}),$$

with $\mathcal{N}$ representing the normal Gaussian probability density, and $\pi_{1,j}$ the mixture proportions (i.e., the probability that a given source belongs to the mixture component $j$).

We obtained maximum-likelihood models using the expectation-maximisation (EM) algorithm for a sequence 