Between Resolution Collapse and Variance Inflation: Weighted Conformal Anomaly Detection in Low-Data Regimes

Most anomaly detection systems output scores rather than calibrated decisions, leaving practitioners to choose thresholds heuristically and without clear statistical interpretation. Conformal anomaly detection addresses this limitation by converting anomaly scores into calibrated p-values that are valid under the statistical assumption of data exchangeability, with a growing literature extending this idea beyond that setting. We present 'nonconform', a Python package for applying conformal anomaly detection within existing machine-learning workflows, and use it as the basis for an implementation-grounded introduction to the field. The package integrates with 'scikit-learn', 'pyod', and custom anomaly detectors, and provides a unified interface for calibration, p-value generation, and false discovery rate control. It supports several conformalization strategies, ranging from simple split-conformal calibration to more data-efficient and shift-aware extensions. Through a progression from foundational concepts to advanced conformalization strategies, complemented by code examples, the paper connects the statistical ideas behind conformal anomaly detection to their practical use in 'nonconform'. Empirical results demonstrate that the implemented methods enable statistically principled anomaly detection. Together, the package and exposition aim to make core conformal anomaly detection workflows more accessible and reproducible in experimental and production-oriented settings.

Conformal prediction (CP) provides powerful, distribution-free prediction sets, but its guarantees rely on the exchangeability of training and test data, which is often violated in practice due to covariate shifts. While weighted conformal prediction (WCP) is designed to handle such shifts, it can suffer from significant undercoverage when the density ratio between the distributions is unbounded and/or must be learned. This is because of both overfitting in learning the density ratio, and high variance in estimating the nonconformity score threshold. To address this, we introduce clipped least-squares importance fitting (CLISF) as a reduced-variance method for density ratio estimation. Specifically, we show that density ratios learned using CLISF, when plugged into WCP, have bounded expected undercoverage. Furthermore, we show that the undercoverage can be corrected by running WCP with a slightly inflated coverage target; crucially, we are able to estimate the required level of inflation from the data. We provide the first theoretical guarantees for weight clipping in conformal inference, achieving dataset-conditional coverage with a sample complexity that does not blow up with the higher moments of the true density ratio -- a key limitation of prior work. We verify our results on real-world benchmarks and synthetic data.

Modern applications of conformal inference to multiple testing problems, such as outlier detection and candidate selection, often involve selecting test samples whose conformal p-values fall below a threshold. The quality of such methods is often measured by the false discovery proportion (FDP), defined as the fraction of incorrect selections. Existing approaches typically control the expected value of the FDP, using methods such as the Benjamini-Hochberg procedure. This approach fails to provide high-probability bounds on the realized false discovery proportion and invalidates statistical guarantees if the rejection threshold is selected after inspecting the data. This paper establishes finite-sample, distribution-free upper bounds on the FDP that hold simultaneously over all possible rejection thresholds, enabling arbitrary post hoc selection of the threshold. Simultaneous validity is achieved by constructing a high-probability envelope for the empirical distribution function of null conformal p-values by sampling from their joint distribution. Furthermore, our framework allows practitioners to modulate the envelope's shape, thereby producing tight bounds in rejection regions of primary interest. We use this flexible approach to derive simultaneous FDP upper bounds for both outlier detection and conformal selection. We demonstrate through synthetic and real-data experiments that the resulting bounds are both valid and substantially less conservative than those derived from existing approaches.