If the model is misspecified, the MLE of the parameter of the misspecified model asymptotically corresponds to the value of the parameter that brings the misspecified model the closest to the true model in the Kullback-Leibler divergence sense. In this respect, the divergence can certainly be made arbitrarily large.

]]>What is meant here is that “even if your model is close in TV (Total Variation) metric to the data generating distribution the output posterior value can be as far as possible from the quantity to be estimated” which is a different statement than “it is difficult for the model to be close (in TV metric, etc…)”.

Concerning the three questions:

(a) Does it matter that the model is misspecified?

The answer is yes because if your model is misspecified and not close in Kullback Leibler divergence (which is a much stronger norm than TV) then the output posterior value can be anything

between the essential min and max of the quantity of interest, i.e. the estimation has no guarantee of accuracy whatsoever.

(b) If it does, is there any meaning in estimating parameters without a model?

Yes, this will be the subject of our sequel work, the idea is to “compute” an optimal formula of the data given the available information. The paper is a bit long and technical because we are also laying down the foundations for such computations.

(c) For a finite sample size, should we at all bother that the model is not “right” or “close enough” if discrepancies cannot be detected at this precision level?

Even if the distance between the model and the data generating distribution is arbitrarily small there is no improvement on the robustness of the estimation.

Houman

]]>http://www.princeton.edu/ ~ umueller / sandwich.pdf

I think it is a great contribution, obtaining a correction analogous to the quasi-maximum likelihood in Bayesian procedures. ]]>

I’ve got to say, I got lost on this paper almost immediately! And, at <80 pages, I wasn't that motivated to find my way again…

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