approach is a good starting point for the complex distribution of field sources shown in Fig. \[data\], for the cluster likelihood model we chose a different class of models in which the probability density of the data in the magnitude-colour subspace of the representation space can be modelled by a curve. In this section, we construct a simplified model of the distribution of cluster members in which the variables pertaining to the proper motions and colour-magnitude subspaces are independent. Therefore, the full likelihood model is decomposed as the product of two models, one for each subspace. The model for the subspace of proper motions still is a mixture-of-Gaussians, while the likelihood model for the colour-magnitude (hereafter CM) subspace is a curvilinear one based on principal curves, as described below. Formally,

$$p(\vec{m}|{\cal C}_2)= \sum_{j=1}^{k_2} \pi_{2,j} \cdot {\cal
  N}_{\vec{\mu}_{2,j},\Sigma_{2,j}}(\mu_{\alpha},\mu_{\delta}) \cdot
p(\vec{m_{CM}}|{\cal C}_2),$$

where $p(\vec{m}_{CM}|{\cal C}_2)$ is the curvilinear likelihood model for the probability density distribution in the CM subspace.

The choice of an empirical curve over a theoretical isochrone frees us from any assumptions on the cluster’s parameters (e.g. age, distance, metallicity) or from the model uncertainties, which are especially important at young ages. We have also tested a single mixture-of-Gaussians model 