In Bayesian inference, we have a dataset $x$ and assumed to come from a known parameterized distribution with unknown parameters $\theta$. We then seek to maximize the posterior $P(\theta|x)$ in order to estimate $\theta$. 

Here, $x$ is treated as fixed. How can we study the robustness of the MAP estimator, $\theta^* = \text{argmax} P(\theta|x)$? I.e. how is the distribution/variance of $x$ used to make statements about how $\theta^*$ changes under different samplings of $x$? I know $\text{Var}(P(\theta|x))$ tells us how "unsure" we are of $\theta^*$ for example. 

More specifically, the [influence function][1] is a measure of robustness of an estimator. How can this be applied to the posterior? 


  [1]: https://en.wikipedia.org/wiki/Robust_statistics#Empirical_influence_function