## rate of convergence for ABC

**B**arber, Voss, and Webster recently posted and arXived a paper entitled *The Rate of Convergence for Approximate Bayesian Computation*. The paper is essentially theoretical and establishes the optimal rate of convergence of the MSE—for approximating a posterior moment—at a rate of 2/(q+4), where q is the dimension of the summary statistic, associated with an optimal tolerance in n^{-1/4}. I was first surprised at the role of the dimension of the summary statistic, but rationalised it as being the dimension where the non-parametric estimation takes place. I may have read the paper too quickly as I did not spot any link with earlier convergence results found in the literature: for instance, Blum (2010, JASA) links ABC with standard kernel density non-parametric estimation and find a tolerance (bandwidth) of order n^{-1/q+4} and an MSE of order 2/(q+4) as well. Similarly, Biau et al. (2013, Annales de l’IHP) obtain precise convergence rates for ABC interpreted as a k-nearest-neighbour estimator. And, as already discussed at length on this blog, Fearnhead and Prangle (2012, JRSS Series B) derive rates similar to Blum’s with a tolerance of order n^{-1/q+4} for the regular ABC and of order n^{-1/q+2} for the noisy ABC…

November 19, 2013 at 11:35 am

To add to the list – Calvet and Czellar (2012, unpublished but available from their websites) derive similar convergence rates for ABC filtering.

November 19, 2013 at 7:49 am

The paper is also available on arXiv: http://arxiv.org/abs/1311.2038

November 19, 2013 at 7:20 am

Dan: yes, 1/q+4 means 1/(q+4)… putting parentheses has a poor visual impact as the lines get apart! As for optimality, I presume it first depends on how you define optimality.

November 19, 2013 at 3:04 pm

True. But I am curious if there is a definition of optimality for which it’s tractable. I mean, we know the minimax rates for posterior contractions (at least for regular cases), but this is a horse of a different colour. I wonder if anyone’s looked at it. Or, in fact, if it’s an unexciting question….

November 19, 2013 at 5:18 pm

I will ask Judith. If someone knows, she is the One..!

November 19, 2013 at 1:49 am

I think there’s a bracket error in the later rates 1/(q+4) etc.

Has there been any work done on optimal rates? Will it always be O(1/q) or is it possible to construct a higher order scheme using the same “replace the data with a summary” idea?