detailed resolution of the real shape, and in both cases the membership probability is defined as the posterior probability of the class given the data in this simplified model. Both works adopt a Bayesian approach to inferring membership probabilities, only in the sense that they apply Bayes’ theorem. With Bayes’ theorem, the posterior probability of the unknown parameter ${\cal C}$ (the class of the source, i.e., member of the cluster or field populations) given the observed data ${\cal D}$ is expressed in terms of a model of the probabilistic distribution of sources of each class in the space of the observed data ${\cal D}$, an [*a priori*]{} membership probability to each class, and a normalising constant:

$$p({\cal C}|{\cal D}) = \frac{p({\cal D}|{\cal C})\cdot p({\cal
    C})}{p({\cal D})}.
\label{bayes.th}$$

We use ${\cal C}$, a dichotomic random variable, to represent membership to the foreground and background populations (${\cal
  C}_1$) or to the stellar cluster (${\cal C}_2$). In the following, the subscript $i$ indexes these two populations, and we use ${\cal
  C}_i$ as an abbreviation of ${\cal C}={\cal C}_i$.

In Eq. \[bayes.th\], the posterior membership probability (the left-hand side) is expressed in terms of the likelihood ($p(D|{\cal
  C})$) and the prior probabilities ($p({\cal C})$ 