How to compare posterior distributions for different observed data? KL-divergence?

So I'm solving an inverse problem with the Bayesian approach $p(u | y) \propto p(y| u )p(u)$.

Assuming I have two datasets $y_1$ and $y_2$, what can be said about the difference in the posteriors $p(u | y_1)$ and $p(u | y_2)$? Can I use Kullback-Leibler divergence for this? (the datasets are assumed to be samples from the same probability distribution)

Also, is there a name for this kind of error type in Bayesian modeling? I quess this problem must be explained somewhere but it seems I'm not searching for the right expressions...

• In what way is this "inverse"? – Sean Easter Mar 23 '16 at 13:58
• It is inverse because there is a relation $y = G(u)$ between the data $y$ and the parameter $u$ and I'm estimating the parameter from the data. It maybe does not matter for the question if the problem is inverse, though – jenna Mar 23 '16 at 14:51
• I've edited your title to better reflect the question, please feel free to roll it back or change it if this is inaccurate. – Sean Easter Mar 23 '16 at 14:55
• Also, you may want to read up on sensitivity analysis and its use in Bayesian methods. (I've usually heard the term refer to analyzing different priors, but you might find the concepts helpful.) – Sean Easter Mar 23 '16 at 14:59
• I've come across it, but so far all I saw in this context was along the lines of different priors or different likelihoods (for the same data). – jenna Mar 23 '16 at 15:07