A non-negligible formula_66 comes with the price that one samples from formula_83 instead of the true posterior formula_6. With a sufficiently small tolerance, and a sensible distance measure, the resulting distribution formula_83 should often approximate the actual target distribution formula_6 reasonably well. On the other hand, a tolerance that is large enough that every point in the parameter space becomes accepted will yield a replica of the prior distribution. There are empirical studies of the difference between formula_83 and formula_6 as a function of formula_66, and theoretical results for an upper formula_66-dependent bound for the error in parameter estimates. The accuracy of the posterior (defined as the expected quadratic loss) delivered by ABC as a function of formula_66 has also been investigated. However, the convergence of the distributions when formula_66 approaches zero, and how it depends on the distance measure used, is an important topic that has yet to be investigated in greater detail. In particular, it remains difficult to disentangle errors introduced by this approximation from errors due to model mis-specification.