  $\alpha$-posteriors and their variational approximations distort standard
posterior inference by downweighting the likelihood and introducing variational
approximation errors. We show that such distortions, if tuned appropriately,
reduce the Kullback-Leibler (KL) divergence from the true, but perhaps
infeasible, posterior distribution when there is potential parametric model
misspecification. To make this point, we derive a Bernstein-von Mises theorem
showing convergence in total variation distance of $\alpha$-posteriors and
their variational approximations to limiting Gaussian distributions. We use
these distributions to evaluate the KL divergence between true and reported
posteriors. We show this divergence is minimized by choosing $\alpha$ strictly
smaller than one, assuming there is a vanishingly small probability of model
misspecification. The optimized value becomes smaller as the the
misspecification becomes more severe. The optimized KL divergence increases
logarithmically in the degree of misspecification and not linearly as with the
usual posterior.
