  Extracting low-dimensional summary statistics from large datasets is
essential for efficient (likelihood-free) inference. We characterise different
classes of summaries and demonstrate their importance for correctly analysing
dimensionality reduction algorithms. We propose obtaining summaries by
minimising the expected posterior entropy (EPE) under the prior predictive
distribution of the model. Many existing methods are equivalent to or are
special or limiting cases of minimising the EPE. We develop a method to obtain
high-fidelity summaries that minimise the EPE; we apply it to benchmark and
real-world examples. We both offer a unifying perspective for obtaining
informative summaries and provide concrete recommendations for practitioners.
