ABC: Why not use the distance measure as a pseudo-likelihood instead? I've read about the ABC rejection algorithm when not being able to calculate the likelihood directly, and my question is: if we have to introduce a distance measure $\rho(D,D')$ anyways, why not use that measure as a pseudo-likelihood to weight the $\theta$ that generated $D'$ instead of thresholding on an arbitrary value of $\epsilon$?
It seems like this would be much more efficient in high dimensional data spaces where you are unlikely to 'hit' close to your original dataset very often.
I realize that this approach is close (identical?) to assuming some measurement error model ($\rho$) for the likelihood. But is that really worse than the  approximation error from the thresholding approach?
 A: This idea has in been implemented in several papers.  Richard Wilkinson's SAGMB paper of 2013 explores the topic in some detail and makes precise the link to assuming a measurement error model.
It turns out to be useful to introduce a parameter $\epsilon$ to the weight function which corresponds to the scale of measurement error.  This operates similarly to the standard ABC threshold; if taken too small the algorithm is very inefficient, but if too large the approximation is poor.
Once this parameter has been introduced, it's not clear whether weighting the simulations is better or worse than thresholding.  In my experience the difference between the two is minimal, especially compared to the effect of other tuning choices, such as $\epsilon$ and the summary statistics.  However, continuous weights are an advantageous feature in some algorithms, for example preventing particle degeneracy in ABC filtering algorithms and allowing emulation of the ABC likelihood.
A: I think that modelling the distribution of the summary statistics is preferable to thresholding, as long as you are able to find a good candidate distribution. For example here the author uses a multivariate normal approximation, which is justified by the asymptotic normality of the statistics. The same method, Synthetic Likelihood, has been used here were in one case 112 summary statistics were used. I don't think you could use that many statistics without making a parametric assumption on the distribution of the statistics. Other advantages of the multivariate normal approximation is that the correlations and scales of the statistics are automatically taken into account by the covariance matrix and that you don't have to choose the tolerance.
Disclosure: I am collaborating with the authors of both papers, so I might be biased.
