We know that given a model or parametric probability distribution say $f(\theta_1, \theta_2, ..., \theta_k)$, it is possible to estimate the $\theta_{i= 1, 2, ..k}$ from the statistics of the sample.
We can use factorization technique or similar other techniques to estimate sufficient, best, complete, etc. kind of statistics.
Is it possible to ask the reverse question, given the statistics say $S_1, S_2,.. S_p$- what is the most general model that could generate the given statistics.
I asked this question to my Professor and didn't get any pointers from her. Does there exist any textbook or research in this direction.