<p>Extracting low-dimensional summary statistics from large datasets is essential for efficient (likelihood-free) inference. We characterize three different classes of summaries and demonstrate their importance for correctly analyzing dimensionality reduction algorithms. We demonstrate that minimizing the expected posterior entropy (EPE) under the prior predictive distribution of the model provides a unifying principle that subsumes many existing methods; they are shown to be equivalent to, or special or limiting cases of, minimizing the EPE. We offer a unifying framework for obtaining informative summaries and propose a practical method using conditional density estimation to learn high-fidelity summaries automatically. We evaluate this approach on diverse problems, including a challenging benchmark model with a multi-modal posterior, a population genetics model, and a dynamic network model of growing trees. The results show that EPE-minimizing summaries can lead to posterior inference that is competitive with, and in some cases superior to, dedicated likelihood-based approaches, providing a powerful and general tool for practitioners.</p>