Specifically, a parametric model is a family of probability distributions that has a finite number of parameters.
A statistical model is a collection of probability distributions on some sample space.
A parametric model is called identifiable if the mapping θ ↦ Pθ is invertible, i.e. there are no two different parameter values θ1 and θ2 such that Pθ1 = Pθ2.
The distinction between these four classes is as follows:[citation needed] Some statisticians believe that the concepts "parametric", "non-parametric", and "semi-parametric" are ambiguous.
[1] It can also be noted that the set of all probability measures has cardinality of continuum, and therefore it is possible to parametrize any model at all by a single number in (0,1) interval.