Then you require a hierarchial model which first estimates the membership of observations to discrete and non-overlapping sets of objects, and second estimates

Then you require a hierarchial model which first estimates the membership of observations to discrete and non-overlapping sets of objects, and second estimates the quantity of interest. This is still a mixture distribution. Statisticians are known to use this kind of model. See https://en.wikipedia.org/wiki/Dirichlet_process https://en.wikipedia.org/wiki/Dirichlet_process