Bayesian methods assume a prior distribution with several hyperparameters. Unfortunately, this is asymptotically incorrect, because distributions in the real world are never exact. For example, the output of flipping a coin doesn't have a Bernoulli distribution, because it may stand on its side; the measured weight distribution of people within a certain can't be normal, because numbers from a normal distribution are "almost surely" irrational.
I propose to assume a distribution of distributions instead of a prior distribution, and then to update the distribution of distributions with sample data as usual. This is a generalized Bayesian method since I'm essentially using $\aleph$ hyperparameters. I suppose doing so would result in an asymptotically correct inference method.
However, applying this method is nontrivial for me. I can't even figure out what should a "distribution of distributions" look like, and updating this distribution is presumably harder. Is there any prior work on this or a similar method?
