and $p({\cal
  D})$). It is precisely the likelihood that is modelled in these pioneering works as a symmetric Gaussian distribution in ${\cal D}$, the space of proper motions.

[@1995AJ....109..672K] combined celestial coordinates with proper motions assuming conditional independency to refine their membership probabilities. They followed the conceptual framework set up by the earlier works cited in the previous paragraph, based on the posterior probability density distribution of a bi-variate Gaussian mixture model. Additionally and alternatively, they proposed an improvement of an original procedure introduced in , where the modelling procedure was treated by partitioning the data into moving bins in a data space that comprises celestial coordinates, proper motions, and the B magnitude. Within each bin, the field distribution is assumed to be a flat linear function in the space of proper motions. The membership probability is then derived using Bayes’ theorem locally for each target star in each bin, fixing the mean of the Gaussian distribution in the vector point diagram (VPD) and fitting the prior probabilities with the least-squares estimate in a 8$\sigma\times8\sigma$ square (with $\sigma$ being the fixed standard deviation of the Gaussian distribution). We recall here that the least-squares fit is equivalent to the maximum-likelihood solution 