What is the common criterion to decide the performance of prior selection in MCMC

Question

For a model with likelihood $p(Y|\theta)$, in which $Y$ is the data and $\theta$ is the parameters. Based on Bayes Rule, we have the posterior

$p(\theta|Y) \propto p(Y|\theta) p(\theta)$

My question is as follows: if we have different prior distributions on $\theta$, e.g., $p(\theta)$ can be a non-informative prior $p(\theta)\propto 1$ or a Gaussian prior $p(\theta) \propto \exp(-\frac{\theta^2}{2})$, if we run a random walk Metropolis Hasting algorithm aiming at posteriors based on different priors, we have one chain for each, then how can we compare the influence of these priors on the posteriors, when considering the MCMC samples?

Answer 1

Bayesian theory suggests that you compute a Bayes factor comparing both [Bayesian] models through the marginal likelihoods of the data $$\mathfrak{B}_{12}(x)=\dfrac{\int f(x|\theta)\,\pi_1(\theta)\,\text{d}\theta}{\int f(x|\theta)\,\pi_2(\theta)\,\text{d}\theta}$$ This may actually be one of the most legitimate usages of the Bayes factor since the priors are then the object of interest, rather than an entry difficult to calibrate.

There are two caveats though:

1. the Bayes factor is not properly defined when one or both of the priors are improper (this is connected with the Jeffreys-Lindley paradox);
2. the computation of the Bayes factor is delicate, although this may be the most favourable setting since you can use bridge sampling there. Along with many other methods.