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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