Anomaly detection using surprisals

Anomaly detection methods are widely used but often rely on ad hoc rules or strong assumptions, and they often focus on tail events, missing ``inlier'' anomalies that occur in low-density gaps between modes. We propose a unified framework that defines an anomaly as an observation with unusually low probability under a (possibly misspecified) model. For each observation we compute its surprisal (the negative log generalized density) and define an anomaly score as the probability of a surprisal at least as large as that observed. This reduces anomaly detection for complex univariate or multivariate data to estimating the upper tail of a univariate surprisal distribution. We develop two model-robust estimators of these tail probabilities: an empirical estimator based on the observed surprisal distribution and an extreme-value estimator that fits a Generalized Pareto Distribution above a high threshold. For the empirical method we give conditions under which tail ordering is preserved and derive finite-sample confidence guarantees via the Dvoretzky--Kiefer--Wolfowitz inequality. For the GPD method we establish broad tail conditions ensuring classical extreme-value behavior. Simulations and applications to French mortality and Test-cricket data show the approach remains effective under substantial model misspecification.