## A novel approach to likelihood-free inference: just use the likelihood!

A new arXival describes “a novel inference framework based on Approximate Bayesian Computation” for a modelling exercise in the field of strong gravitational lensing.  Since they acknowledge insightful discussions from the guys who seem to deliberate refuse to cite my work on ABC for astronomy, it’s no surprise that the statistical analysis again goes astray.  Basically, the proposal here is to use the negative log marginal likelihood—the marginalisation being over a set of nuisance parameters conditional on the observed data and a given set of hyper-parameters—as a distance in ABC.  Usually ABC is motivated because the likelihood is not available, but that doesn’t seem to be understood here.  The only thing that seems to stop the posterior collapsing to the mode is the early stopping of the ABC-SMC algorithm at an arbitrary number of steps.  There’s also some silliness with respect to the choice of prior for the source image (the component of the model that is marginalised out exactly) which is emphasised in the paper as disfavouring realistic source brightness distributions.

The only thing I found interesting here was the reference to Vegetti & Koopmans (2009)  [Note: this is clearly the paper the authors intended to cite, rather than the other Vegetti & Koopmans 2009: MNRAS 400 1583; this kind of mistake suggests the level of care going into these arXivals is less even than the few minutes I spend cranking out a blog post].  The Vegetti & Koopmans (2009) method involves construction of a source image prior via a Voronoi tessellation scheme with regularisation terms.  An interesting project would be to examine how the SPDE approach could allow for a more nuanced prior choice, introduction of non-stationarity etc (see Lindgren et al. 2011).

Actually there is another interesting thing that could be investigated with this model.  The authors choose to look at $-2 \log P(d|\theta)$ for their distance: the factor of 2 is of course irrelevant in the ABC context and with respect to collapse on the posterior mode; but for likelihood-based inference it would represent an extra-Bayesian calibration factor, which could actually be chosen to reduce exposure to excess concentration in the mis-specified setting via e.g. the loss-likelihood bootstrap.

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