I understand that the Jeffreys prior provides a method for constructing a prior distribution over parameters for a given model (likelihood function) such that the prior distribution is "invariant under reparameterization." I understand this invariance to mean that the Jeffreys prior for a given set of parameters can be converted to a prior distribution over a second set of parameters (via the standard change of variables method for probability distributions), and the resulting prior will match the Jeffreys prior for the second set of parameters.
Does a similar kind of invariance exist for:
- the posterior distribution based on the Jeffreys prior? (i.e. does the posterior derived from a Jeffreys prior possess the same invariance properties as the Jeffreys prior?)
- the prior predictive distribution based on the Jeffreys prior? (i.e. does the prior predictive distribution derived from a set of parameters and the corresponding Jeffreys prior match the prior predictive distribution derived from a second set of parameters and the corresponding Jeffreys prior?)
- the posterior predictive distribution based on the Jeffreys prior? (similar to #2)
- the MAP (maximum a posteriori) parameter estimate based on the Jeffreys prior? (i.e. does the distribution over the data space based on one set of MAP parameters match the distribution based on the MAP parameters for a different parameterization, where both parameter posterior distributions are based on the corresponding Jeffreys priors)