I don't know Bayesian statistics very well, so I don't know if the question makes sense. Let me give an example.
We assume that the income distribution of a country is a Pareto distribution (the minimum is C, and the shape alpha is 3). Now we want to estimate the income distribution of a specific company (it is inside this country). We assume that it is also a Pareto distribution and we have 100 observations.
With frequentist consideration, we can fit a Pareto distribution with these 100 observations (with MME or MLE). But I am wondering if it is possible to estimate with the prior distribution, which is the income distribution of a whole country.
And we consider that the minimum is the same, but we want to estimate the new shape alpha'.
Is it possible to consider that the prior distribution is the Pareto distribution (of the country, with known C and alpha) and after updating (with the 100 observations), we still get a posterior distribution that is a Pareto (with C, the same minimum and a new alpha)?