Robustness of prior distribution?

Dear Johannes,
Thank you for your help last time, I am grateful.
I have two questions:
First, in Bayesian estimation of DSGE models, is significant difference between posterior mean/median and prior distribution for most parameters better than insignificant difference between posterior mean/median and prior distribution?
Second, does significant difference between posterior mean/median and prior distribution indicate robustness of prior distribution?
Thank you very much and look forward to hearing from you.
Best regards,
PhD Candidate

Generally, if the prior and the posterior coincide, this is a sign of trouble. It could happen because you indeed picked the true distribution of the parameters already as your prior, so there is no point in updating it based on the data. But this pretty much never happens. So if the two coincide, the data is not informative for updating your prior distribution (one of the reasons this may happen is if parameters are not identified in the data, so any data would not help). For that reason, if your posterior looks different, that is generally a good sign. However, you cannot say anything about this being robust to changing the prior. If you were to use a very tight and informative prior, it would most probably still dominate the data. Put differently, just because the posterior as the weighted average of prior and likelihood suggests that the likelihood is different from the prior, does not mean that this weighted average will be the same for whatever the prior is.