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Extracting low-dimensional summary statistics from large datasets isessential for efficient (likelihood-free) inference. We characterise differentclasses of summaries and demonstrate their importance for correctly analysingdimensionality reduction algorithms. We propose obtaining summaries byminimising the expected posterior entropy (EPE) under the prior predictivedistribution of the model. Many existing methods are equivalent to or arespecial or limiting cases of minimising the EPE. We develop a method to obtainhigh-fidelity summaries that minimise the EPE; we apply it to benchmark andreal-world examples. We both offer a unifying perspective for obtaininginformative summaries and provide concrete recommendations for practitioners.

Minimising the Expected Posterior Entropy Yields Optimal Summary Statistics