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?