Leakage Impact Score: Measuring Information Leakage

• Leakage Impact Score (LIS) is a family of quantitative indices designed to measure and benchmark the extent and severity of information leakage in statistical, predictive, and market-based models.
• It leverages formal definitions—such as pre-news price moves, conditional mutual information, and paired performance differences—to assess leakage and ensure reproducibility and auditability.
• Its applications span prediction markets and machine learning models, where rigorous timestamping, anchor sensitivity, and edge constraints are critical for accurate leakage detection.

The Leakage Impact Score (LIS) is a family of quantitative indices devised to measure, benchmark, and detect the extent, severity, or practical impact of information leakage in statistical, predictive, and market-based models. While the “LIS” designation appears across several research subfields, underlying all its variants is the formalization and operationalization of how extra-systemic or target-leaking information enters and inflates apparent predictive performance or economic profit. LIS has emerged as a central criterion for reproducibility, auditability, and regulatory compliance in contemporary data science and prediction-market analysis.

1. Formal Definitions and Mathematical Foundations

1.1 Information Leakage Score in Prediction Markets

The primary LIS formalism for event-resolved binary prediction markets is defined as the proportion of a market’s terminal information move that is “front-loaded,” i.e., realized in prices prior to the first public news event. For a resolved binary market $M$ with time-of-first-trade $T_\mathrm{open}$, news timestamp $T_\mathrm{news}$, resolution time $T_\mathrm{res}$, mid-price process $p(t)$, and realized binary outcome $p_\ast \in \{0,1\}$, the Information Leakage Score is:

$$\mathrm{ILS}(M) = \frac{p(T_\mathrm{news}) - p(T_\mathrm{open})}{p_\ast - p(T_\mathrm{open})}.$$

$\mathrm{ILS}(M) = 1$ implies the entire move was priced in pre-news; $=0$ means no pre-news drift; values $>1$ or $T_\mathrm{open}$0 indicate overshoot or counter-drift (Nechepurenko, 1 May 2026).

1.2 Deadline-Resolved Variants

To handle deadline markets—dominant in documented insider cases—a “deadline-ILS” extension ($T_\mathrm{open}$1) is defined. For deadline $T_\mathrm{open}$2 and event time $T_\mathrm{open}$3:

$T_\mathrm{open}$4

$T_\mathrm{open}$5

Anchor choice, hazard modeling, and outcome flipping for $T_\mathrm{open}$6 are specified for category-specific application (Nechepurenko, 4 May 2026).

1.3 Information-Theoretic Leakage Impact in Concept Models

For concept bottleneck models (CBMs), the LIS is defined via conditional mutual information:

$T_\mathrm{open}$7

where $T_\mathrm{open}$8 are input features, $T_\mathrm{open}$9 ground-truth concepts, $T_\mathrm{news}$0 the bottleneck embedding, and $T_\mathrm{news}$1 the target. It quantifies the reduction in prediction uncertainty for $T_\mathrm{news}$2 due to information in $T_\mathrm{news}$3 not accounted for by $T_\mathrm{news}$4 alone (Makonnen et al., 13 Apr 2025).

1.4 Empirical “Paired-Delta” Leakage Impact

In cross-validation and tabular data settings, LIS aggregates raw paired performance differences (e.g., mean $T_\mathrm{news}$5AUC) and standardized effect size (Cohen’s $T_\mathrm{news}$6) across leakage classes:

$T_\mathrm{news}$7

with referencing to negligible-noise benchmarks $T_\mathrm{news}$8 and $T_\mathrm{news}$9 (Roth, 5 Apr 2026).

2. Operational Preconditions and Scope Criteria

All formal LIS variants are only interpretable under strict operational constraints.

• Edge-effect condition: For prediction-market scores, $T_\mathrm{res}$0; i.e., avoid trivial denominators.
• Non-trivial total move: $T_\mathrm{res}$1 (empirically $T_\mathrm{res}$2 suffices).
• Anchor sensitivity: Robustness of LIS under several plausible timestamp choices must be checked; significant qualitative shifts under minor anchor perturbations invalidate results.
• Resolution Typology: Markets must be classified as event-resolved vs deadline-resolved; each requires a tailored LIS.
• Price/proxy quality: For market LIS, quality and provenance of $T_\mathrm{res}$3 or $T_\mathrm{res}$4 critically affect interpretability.

These criteria collectively prevent spurious LIS elevation due to price pathologies, anchoring artefacts, or misclassification. The same logic applies to concept models if ground-truth concept sufficiency is in doubt (Nechepurenko, 1 May 2026, Nechepurenko, 4 May 2026, Makonnen et al., 13 Apr 2025).

3. Statistical Interpretation and Decomposition

3.1 Murphy-Decomposition Link

For prediction markets, $T_\mathrm{res}$5 admits a decomposition in terms of the Brier score and its Murphy decomposition:

$T_\mathrm{res}$6

where ILS approximates the ratio $T_\mathrm{res}$7: the fraction of total outcome resolution front-loaded before the news, linking LIS directly to the literature on proper scoring rules and probabilistic calibration (Nechepurenko, 1 May 2026).

3.2 Information-Theoretic View

For CBMs, $T_\mathrm{res}$8 is interpreted as the quantification of “shortcut learning”: the extend to which the bottleneck encoding $T_\mathrm{res}$9 contains predictive information about $p(t)$0 not mediated by the intended intermediate variables $p(t)$1 (Makonnen et al., 13 Apr 2025).

3.3 Empirical Delta Interpretation

For tabular and temporal ML, LIS establishes a unified scale for cross-paper comparability by referencing a background noise floor for various leakage mechanisms, converting diverse effect sizes ($p(t)$2AUC, $p(t)$3) into a dimensionless severity scale (Roth, 5 Apr 2026).

4. Empirical Results and Pilot Findings

Studies report that naive or poorly anchored LIS computations can yield misleading conclusions:

• Using simple time proxies for $p(t)$4 in market LIS fails to separate known insider-trading domains from controls; true article-derived anchors are indispensable (Nechepurenko, 1 May 2026).
• In the 2026 Polymarket U.S.–Iran contracts, deadline-ILS with article-derived event anchors yielded ILS$p(t)$5 vs $p(t)$6 for legacy proxies, a 0.444 divergence and reversal of sign (Nechepurenko, 4 May 2026).
• In synthetic CBM experiments, LIS declines monotonically as true leakage is reduced, validating faithfulness for $p(t)$7, well-calibrated probability estimates, and classifier choice. XGBoost is empirically superior for entropy approximation, with Random Forest yielding instability in low-$p(t)$8 (Makonnen et al., 13 Apr 2025).
• Across 2,047 datasets, estimation leakage (normalization on full data) yields negligible $p(t)$9, while selection and memorization leakages routinely reach $p_\ast \in \{0,1\}$0 (Roth, 5 Apr 2026).

5. Detection Floor, Limits, and Auditable Protocols

Recent decision-theoretic analysis establishes a detection floor for output-only LIS diagnostics. For any calibrated leak that does not increase discrimination (e.g., measured by Harrell’s concordance $p_\ast \in \{0,1\}$1) beyond an “honest” baseline by $p_\ast \in \{0,1\}$2, leakage becomes undetectable from predictions and outcomes alone:

$p_\ast \in \{0,1\}$3

For the UK Biobank delirium endpoint, $p_\ast \in \{0,1\}$4—below this, all smooth, well-calibrated leakage is observationally indistinguishable from genuinely improved models. Only near-deterministic leaks (unit-purity head events) or broad discrimination gains produce a detectable LIS trigger (Jacobs, 9 Jun 2026).

Detection algorithms operate by evaluating concordance, cumulative purity curves, and a mismatch to a known “honest” reference; decision rules return “leaky” only if the spike or breadth criteria are met. Otherwise, residual leakage must be considered below material impact and undetectable in the absence of process code, training data, or external discrimination ceilings.

6. Best Practices and Practical Guidance

6.1 Market and Financial Applications

• Collect and curate first-trade, public-news, and article-derived timestamps for all studied contracts.
• For deadline contracts, fit per-category exponential hazard rates for expected event times; avoid applying a single model to heterogenous categories.
• Perform anchor-sensitivity analyses and flag any case where LIS sign or order-of-magnitude is not robust.
• Enforce edge, total-move, and typology constraints before interpreting LIS for market surveillance, compliance, or insider trading audits (Nechepurenko, 1 May 2026, Nechepurenko, 4 May 2026).

6.2 Statistical and ML Benchmarks

• Report $p_\ast \in \{0,1\}$5AUC, $p_\ast \in \{0,1\}$6, and LIS for each leakage class; contextualize scores via noise floor benchmarks.
• Use well-calibrated, high-capacity estimators (XGBoost with temperature scaling) for mutual-information-based LIS estimations in CBMs.
• For output-only auditing, explicitly reference a discrimination ceiling or honest model, and interpret sub-threshold increments as non-actionable.
• Publish per-dataset, per-class LIS streams and the codebase for metric computation to facilitate transparent audit and downstream meta-analyses (Makonnen et al., 13 Apr 2025, Roth, 5 Apr 2026, Jacobs, 9 Jun 2026).

7. Methodological Extensions and Open Considerations

The LIS methodology continues to evolve in response to complex leakage scenarios and practical limitations:

• Deadline-ILS and exponential-hazard anchoring close methodological gaps in prediction-market leakage detection (Nechepurenko, 4 May 2026).
• Prior-free detection identifies limits of output-only LIS for identifying “hard-to-detect” leakages, necessitating domain priors, audit trails, or process-level artifacts for full assurance (Jacobs, 9 Jun 2026).
• Cross-market wallet analysis and continuous price and trade collection are prioritized infrastructure for correlating LIS with trader coordination in financial systems.

A plausible implication is that as LIS becomes embedded in reproducibility and regulatory standards, formalization of scope criteria, anchor fidelity, and reference model establishment will be central in both scientific and applied audit workflows.

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Key References:

LIS Variant	Field	Key Paper
Prediction Market ILS / ILS$p_\ast \in \{0,1\}$7	Financial/Market Analysis	(Nechepurenko, 1 May 2026, Nechepurenko, 4 May 2026)
Information-Theoretic LIS	Concept Bottleneck Models	(Makonnen et al., 13 Apr 2025)
Output-Only Detection Floor	General ML Auditing	(Jacobs, 9 Jun 2026)
Unified Empirical (ΔAUC/dz) LIS	ML Benchmarking	(Roth, 5 Apr 2026)