## model misspecification in ABC

**W**ith David Frazier and Judith Rousseau, we just arXived a paper studying the impact of a misspecified model on the outcome of an ABC run. This is a question that naturally arises when using ABC, but that has been not directly covered in the literature apart from a recently arXived paper by James Ridgway [that was earlier this month commented on the ‘Og]. On the one hand, ABC can be seen as a robust method in that it focus on the aspects of the assumed model that are translated by the [insufficient] summary statistics and their expectation. And nothing else. It is thus tolerant of departures from the hypothetical model that [almost] preserve those moments. On the other hand, ABC involves a degree of non-parametric estimation of the intractable likelihood, which may sound even more robust, except that the likelihood is estimated from pseudo-data simulated from the “wrong” model in case of misspecification.

In the paper, we examine how the pseudo-true value of the parameter [that is, the value of the parameter of the misspecified model that comes closest to the generating model in terms of Kullback-Leibler divergence] is asymptotically reached by some ABC algorithms like the ABC accept/reject approach and not by others like the popular linear regression [post-simulation] adjustment. Which suprisingly concentrates posterior mass on a completely different pseudo-true value. Exploiting our recent assessment of ABC convergence for well-specified models, we show the above convergence result for a tolerance sequence that decreases to the minimum possible distance [between the true expectation and the misspecified expectation] at a slow enough rate. Or that the sequence of acceptance probabilities goes to zero at the proper speed. In the case of the regression correction, the pseudo-true value is shifted by a quantity that does not converge to zero, because of the misspecification in the expectation of the summary statistics. This is not immensely surprising but we hence get a very different picture when compared with the well-specified case, when regression corrections bring improvement to the asymptotic behaviour of the ABC estimators. This discrepancy between two versions of ABC can be exploited to seek misspecification diagnoses, e.g. through the acceptance rate versus the tolerance level, or via a comparison of the ABC approximations to the posterior expectations of quantities of interest which should diverge at rate Vn. In both cases, ABC reference tables/learning bases can be exploited to draw and calibrate a comparison with the well-specified case.

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