Today I read an arXival on the topic of data compression or likelihood compression for cosmology, which translates to construction of approximately (or exactly) sufficient summary statistics via a transformation with reference to the assumed likelihood function. As Alsing & Wandelt point out, this type of transformation falls in the class of score function likelihood summaries. Both papers mention the possibility of using these summaries as low dimensional targets for likelihood-free inference methods comparing mock data simulations to the observed cosmological data. In this case the technique becomes the ABC-IS of Gleim & Pigorsch (see also here & here), and has a connection to indirect inference and the method of moments. I’m somewhat skeptical that a realistic application would meet the conditions needed for sufficiency (with respect to the simulation-based model) of the auxiliary summary statistics; but some of the general insights might cross over (e.g. the auxiliary model should aim to be as richly descriptive as the simulation-based model, even if structural/computationally simpler).