It’s true that importance sampling doesn’t require linearity. The problem is that importance sampling fails precisely when the new posterior (with changed hyperparameters) doesn’t overlap much with the original posterior, which is exactly the case you need to know about when robustness is a problem.

In contrast, linearity can hold far beyond the mass of the original posterior. Consider, for example, the sensitivity of a multivariate normal parameter to its prior mean — the dependence is linear for all values of the prior mean, even when there’s no overlap at all and importance sampling fails completely.

With exactly your thought in mind, I actually initially implemented importance sampling as a sanity check for linearity in this package, but took it out of the API for exactly the above reasons. It’s just not very useful in cases of real non-robustness. (I’d be happy to put it back in there’s a demonstrable need!)

Incidentally, the linear approximation we describe is also a linear approximation to the importance sampling estimate — see appendix B of our paper, Covariances, Robustness, and Variational Bayes, for a proof: https://arxiv.org/pdf/1709.02536.pdf

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