under the assumption of independent Gaussian uncertainties with constant standard deviation. Hence the grouping of the data in magnitude bins. This approach (binning in magnitude) became popular in later works [see @2012MNRAS.422.1495L for a recent example]. This latter alternative in the work by [@1995AJ....109..672K] presents several disadvantages in our opinion, including the occasional introduction of faint stars or fractions thereof ([*sic*]{}) to avoid negative values of the field-star distribution.

An alternative to this parametric modelling of the distribution of cluster and field members in the VPD was proposed by , who derived kernel-based estimations of the probability density in the VPD space both for field stars alone (in a region assumed free from cluster members) and for the mixed population in the cluster celestial region.

Later works with combined astrometric and spectro-photometric data sets elaborated further on these early works. Broad-band photometry is usually considered separately from the astrometric measurements, either before or after the astrometric analysis, and as hard selection thresholds in colour-magnitude or colour-colour diagrams (hereafter CMD and CCD respectively; see for example or ). Usually, these thresholds do not incorporate the photometric uncertainties in a way that retains the probabilistic nature of the memberships, and consist of 