We propose a new model-agnostic search strategy for physics beyond the standard model (BSM) at the LHC, based on a novel application of neural density estimation to anomaly detection. Our approach, which we call classifying anomalies through outer density estimation (cathode), assumes the BSM signal is localized in a signal region (defined e.g., using invariant mass). By training a conditional density estimator on a collection of additional features outside the signal region, interpolating it into the signal region, and sampling from it, we produce a collection of events that follow the background model. We can then train a classifier to distinguish the data from the events sampled from the background model, thereby approaching the optimal anomaly detector. Using the LHC Olympics R&D dataset, we demonstrate that cathode nearly saturates the best possible performance, and significantly outperforms other approaches that aim to enhance the bump hunt (cwola hunting and anode). Finally, we demonstrate that cathode is very robust against correlations between the features and maintains nearly optimal performance even in this more challenging setting.