I have two Bayesian regression models: both use the same data $D$ and all same specification $M$ but the different priors $p_1(\theta)$ and $p_2(\theta)$.
Can I interpret the posteriors from the two models, $p(\theta|D,M,p_1(\theta))$ and $p(\theta|D,M,p_2(\theta))$ independently and compare them? One example of comparing the results is something like "one model results in an expected loss of 1 while the other results in an expected loss of 2, so that given the different priors, the range of the expected loss is between 1 and 2." Or do I actually have to condition each model not only on its own prior but also on the prior of the other model, in order to estimate the single posterior that captures the uncertainty of the prior choice, using a hyperprior on the prior choice, such that the posterior to be estimated is $p(\theta|D,M,p(p_i(\theta)))$? My intuition is that interpreting each posterior independently means I implicitly assume that the hyperprior is a uniform distribution.
I have come across this question, which talks about estimating a posterior using all different datasets together while using the same prior. My setting is opposite, using the same dataset while using different priors.
I would appreciate your suggestions.