summary statistics for ABC. Approaches using machine learning tools such as random forests [@pudlo2015reliable] to aid model selection and neural networks to form a parameteric model of the posterior [@papamakarios2016fast; @papamakarios2018sequential] have also been investigated. Other work has avoided using summary statistics at all by considering Wasserstein distance between full data sets [@bernton2017inference].

In this work, we approach the problem from the point of view of finding the right distance function, adapted to information contained in the summary statistics, rather than selecting a certain subset of summary statistics. We provide an automatic algorithm that adaptively selects weights for each summary statistic within the ABC distance function.

Outline {#Outline}
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Our contribution in this work is to present a flexible, novel framework for improving inference with ABC by adapting the weights of different summary statistics to maxmize the gain in posterior information from a dataset. This is helpful for avoiding bias and variance from redundant information in data (such as would be the case when including a summary statistic that is uncorrelated with the parameters of interest). A further advantage of our work is that it can alleviate the burden of designing and selecting summary statistics ‘by hand’, since a large 