stat.MLAug 28, 2026

Localizing Global Discrepancies: Marginal Contributions and Contextual Anomaly Detection

Authors: Tommaso dorigo

Organizations: INFN, Sezione di Padova, Padova, Italy · Department of Computer Science, Electrical and Space Engineering, Luleå University of Technology, Luleå, Sweden

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Abstract

Global goodness-of-fit and discrepancy statistics can establish that a sample departs from a reference distribution without identifying which observations drive the departure. We develop a framework for this localization problem by assigning to each observation its conditional or marginal contribution across random statistical contexts. This connects resampling diagnostics and data valuation to projection theory and event-level anomaly detection. For symmetric statistics, fixed-size replacement is exactly equivalent to centered conditional localization. For U-statistics, the addition score equals the first Hoeffding/Hájek contribution; for smooth distributional functionals it is related at leading order to the influence function; and for unbiased known-background MMD it reduces exactly to the MMD witness. This viewpoint also yields more efficient estimators. Matched-context subtraction removes fluctuations unrelated to the observation, while for pairwise MMD the event-containing terms give a simple localizer. On the LHC Olympics anomaly-detection benchmark, the pair estimator converges to the direct empirical MMD witness with the predicted 1/(Rm^2) scaling, where m is batch size and R the number of batches. At m=1000 and R=5x106 it reaches correlation 0.9993 with essentially identical AUC. We also ask when context contains information beyond an event's own features. In a shared-latent toy model, the full single-event signal and background distributions are identical by construction, forcing isolated-event AUC=0.5. Discriminating information survives only in cross-event dependence induced by the shared latent parameter; the ensemble recovers this information, whereas an independent-latent control does not. This separates two roles of context: efficient localization of a global discrepancy and genuinely additional class information when the alternative contains shared structure.

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Figure 1: Direct validation of conditional MMD localization by explicit resampling. Event scores obtained by forcing individual events into repeatedly resampled batches are compared with the analytic conditional expectation of Eq. ( 71 ). Signal and background events follow the same predicted relation, showing that explicit resampling recovers the conditional projection of the batch statistic rather than defining an independent anomaly score.
Figure 2: Event-level ROC curves for one representative pseudoexperiment in the controlled fixed-alternative study. Ensemble-averaged results are reported in the text.
Figure 3: Dependence of the MMD localization score and raw-resampling uncertainty on batch size mm , for R=3000R=3000 forced-inclusion contexts per probe event. Left: signal–background localization separation; both the analytic prediction and finite-pseudoexperiment result follow the exact 1/m1/m scaling. Right: RMS Monte Carlo uncertainty relative to ΔGm\Delta G_{m} . The relative error initially falls, reaches a minimum near m≃80m\simeq 80 , and then rises. This U-shaped behavior marks the crossover from a near-degenerate second-order regime toward the generic first-order Hoeffding regime.
Figure 4: Monte Carlo efficiency of marginal and event-containing MMD localization in the controlled Gaussian experiment, using R=1000R=1000 forced-inclusion contexts per probe event. Left: RMS score uncertainty divided by the common class separation ΔGm\Delta G_{m} . Raw resampling remains on the near-degenerate plateau, whereas marginal and pair estimators closely follow the m−1/2m^{-1/2} reference. Right: variance ratios relative to raw averaging. They fall approximately as m−1m^{-1} because the raw MMD statistic is still in its O(m−2)O(m^{-2}) near-degenerate regime, while the efficient estimators scale as O(m−3)O(m^{-3}) .
Figure 5: Recovery of collective classification information with ensemble size in the shared-latent experiment. Left: mean AUC for the true- μ\bm{\mu} likelihood-ratio oracle, true- μ\bm{\mu} MMD witness, empirical LOO witness, and independent- μi\bm{\mu}_{i} control. Only the shared-context empirical witness improves systematically with NN . Right: fraction frecf_{\rm rec} of the MMD classification advantage above chance recovered by the empirical contextual witness. Error bars are standard errors across pseudoexperiments.
Figure 6: Recovery of the empirical contextual score by marginal and event-containing estimators for N=5000N=5000 and m=100m=100 . Left: shared- μ\bm{\mu} construction. Both Monte Carlo estimators approach the empirical LOO target ( AUC=0.7584\mathrm{AUC}=0.7584 ); the true- μ\bm{\mu} MMD witness ( AUC=0.8448\mathrm{AUC}=0.8448 ) shows the remaining finite-ensemble loss. Right: independent- μi\bm{\mu}_{i} control. The estimators converge to the accidental finite-sample LOO value ( AUC≃0.5177\mathrm{AUC}\simeq 0.5177 ), while the ensemble-average control remains consistent with chance. The marginal and pair estimators are nearly indistinguishable in both cases.
Figure 7: Raw conditional resampling, matched marginal contributions, and event-containing pair interactions versus batch size at fixed R=106R=10^{6} in the LHC Olympics mock sample. Left: AUC. Right: maximum SIC. The horizontal line is the direct empirical witness on the same mock sample. Increasing mm does not change the population ordering but strongly improves the Monte Carlo precision of marginal and pair estimators.
Figure 8: Convergence with the number of sampled batches at fixed m=1000m=1000 . Left: correlation with the direct empirical witness. Right: maximum SIC. Marginal and pair estimators rapidly approach the witness, whereas the raw batch score remains much noisier. The substantially slower convergence of SIC than of the global score correlation shows that accurate ordering in the extreme tail is more demanding than reproducing the witness score field overall.
Figure 9: Scaling of the pair-interaction estimator. The decorrelation measure qρ=ρ−2−1q_{\rho}=\rho^{-2}-1 is plotted against Rm2Rm^{2} for the different mm and RR configurations. At sufficiently large exposure the points collapse onto a common power law. The dashed line shows the predicted α=1\alpha=1 behavior.
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