Normalized Deviation Faithfulness (NDF)

• Normalized Deviation Faithfulness (NDF) is a quantitative framework that normalizes performance gaps to measure how faithfully a sub-network replicates a full model’s behavior.
• It employs a clipping mechanism to ensure values remain between 0 and 1, addressing numerical instability found in prior faithfulness metrics.
• NDF is applied across language model circuit discovery, probabilistic filtering, and stochastic process analysis to benchmark surrogate fidelity and theoretical consistency.

Normalized Deviation Faithfulness (NDF) is a quantitative framework for assessing how closely a subsystem, surrogate, or representation matches the ground-truth behavior of a reference probabilistic or functional system, with performance deviations normalized to account for intrinsic variability or baseline uncertainty. NDF as a term and as a metric appears in the contemporary literature across several domains—notably in LLM circuit discovery, statistical filtering, and stochastic process theory—where it provides a robust quantitative yardstick for “faithfulness” or “honesty” in probabilistic, generative, or representational modeling. This article synthesizes the technical definitions, theoretical underpinnings, and empirical roles of NDF as established in recent research.

1. Formal Definition in Circuit Discovery

NDF is formally defined in the context of LLM circuit discovery, especially in the analysis of query circuits within LLMs (Wu et al., 29 Sep 2025). Given a reference model $M$, query $q$, a discovered sub-network (circuit) $C_{(q)}$, and a corrupted version of the query $q'$, the NDF of circuit $C_{(q)}$ is:

$$\mathrm{NDF}(C_{(q)}) = 1 - \min\left( \left| \frac{L(M(q)) - L(C_{(q)}(q))}{L(M(q)) - L(M(q'))} \right|, 1 \right)$$

where $L(\cdot)$ is the task-specific performance metric—for example, log-likelihood, accuracy, or a scoring function:

• $L(M(q))$ denotes the model’s score on the original input;
• $L(C_{(q)}(q))$ is the circuit’s score for the same input;
• $L(M(q'))$ is the model’s score on a corrupted “negative” input with crucial cues removed.

This metric normalizes observed deviations: perfect recovery ($L(C_{(q)}(q)) = L(M(q))$) yields $\mathrm{NDF}=1$; no better than the corrupted baseline yields $\mathrm{NDF}=0$. The value is always clipped within $[0,1]$.

The normalization via the performance gap $(L(M(q)) - L(M(q')))$ makes NDF robust to datasets or tasks with small or large intrinsic difficulty, ensuring a stable scale.

2. Theoretical Motivation and Comparison to Prior Metrics

Prior to NDF, several metrics for quantifying faithfulness were employed, such as Normalized Faithfulness Score (NFS). However, NFS was shown to lack numerical stability in practice, sometimes exceeding 1 or taking negative values on natural datasets (e.g., MMLU), especially when the gap between original and corrupted query performance is small.

NDF’s normalization and clipping mechanism guarantees:

• Boundedness: Always within $[0, 1]$, enabling direct interpretability.
• Symmetry: Deviations in either direction are penalized equally.
• Robustness: Remains well-behaved across a diversity of benchmarks and query types.

These mathematical properties are critical for progress-tracking and benchmarking methods that produce sparse, approximate, or modular representations of a model’s internal computations (Wu et al., 29 Sep 2025).

3. Methodological Applications Across Domains

3.1 LLM Circuit Discovery

In the context of query circuit discovery:

• Circuits are identified as minimal sub-networks responsible for answering individual queries.
• NDF is computed per query, providing a faithfulness signal for each sub-network’s recovery of the reference model’s decision.
• Best-of-N and other sparse selection strategies are evaluated using average NDF over benchmarks such as IOI, arithmetic, MMLU, and ARC.
• Empirical findings: circuits utilizing as little as 1.3% of a model’s connections can recover approximately 60% of MMLU performance, corresponding to an average NDF near 0.6.

NDF thus enables direct, quantitative assessment of whether a discovered circuit or compressed model is a faithful proxy for the full model's reasoning on a given input.

3.2 Consistency of Probabilistic Filters

Exact consistency tests for Gaussian mixture filters employ the “normalized deviation squared” (NDS) statistic, which underpins an NDF-like framework for validating estimator faithfulness (Ahmed et al., 2023). The NDS statistic:

$$q(x) = (x - \bar{\mu})^\top \bar{\Sigma}^{-1} (x - \bar{\mu})$$

for mixture mean $\bar{\mu}$ and covariance $\bar{\Sigma}$, generalizes to mixtures of generalized chi-square distributions. Consistency tests aggregate such $q(x)$ over time, comparing observed distributions to theoretical ones, ensuring that uncertainty estimates are “honest” vis-à-vis the true state dynamics.

In this context, normalized deviation (via NDS) quantifies whether the probabilistic estimate is commensurate, thereby establishing NDF in the stochastic filtering regime.

3.3 Large Deviation Principles in Stochastic Processes

The term “normalized deviation faithfulness” is also associated with topology choices in large deviation theory for Lévy processes (Dort et al., 2023). The Skorokhod M1 topology is advocated as NDF: it ensures that large deviation costs (rate functions) are insensitive to extraneous pathwise details (e.g., exact jump timing), instead reflecting the true, normalized cost of pathwise rare events. In coarser topologies (such as M1), the rate function aligns with the natural normalization (e.g., exponent $\alpha'$ of the decay in probability for rare event functionals), yielding exponential tightness and honest tail asymptotics. This establishes NDF as an organizing principle for the topology of function spaces in probability theory, ensuring that large deviation estimates are faithful to actual rare event costs.

4. Empirical and Numerical Validation

NDF’s practical merit is established via empirical results:

• In language modeling, NDF reveals that extremely sparse sub-networks can approximate the decision boundaries of full models with high compressed faithfulness (as tracked by NDF), and that non-selected sub-networks have low NDF, confirming their lack of necessity (Wu et al., 29 Sep 2025).
• In filtering, distributions of NDS statistics match theoretical predictions to high accuracy; when models are calibrated, observed statistics fall within credible intervals, failing which NDF tests flag inconsistency (Ahmed et al., 2023).
• In large deviation theory, empirical tail probabilities for the area and supremum of normalized excursions exhibit precisely the expected logarithmic decay rates, demonstrating that the M1 topology’s adopting of NDF is justified (Dort et al., 2023).

5. Broader Implications and Technical Significance

The introduction and adoption of NDF enable:

• Comparative analysis of sparse or modular representations of function, with robust and interpretable quantification of performance retention.
• Improved reliability in hypothesis testing, consistency validation, or rare event estimation across domains.
• Theoretical progress in topology selection for stochastic process analysis, ensuring model-derived results align with practical, normalized deviation costs.

Furthermore, NDF’s general normalization approach makes it adaptable: it resists artifacts arising from negligible or excessive differences between models, corrupted instances, or baseline references, allowing it to function reliably as a proxy for interpretability, uncertainty calibration, or tail risk quantification across highly disparate settings.

6. Summary Table of NDF Frameworks

Domain	Core NDF Statistic/Formula	Interpretive Role
LLM Circuits (Wu et al., 29 Sep 2025)	$$\mathrm{NDF}(C_{(q)}) = 1 - \min\left( \left| \frac{L(M(q)) - L(C_{(q)}(q))}{L(M(q)) - L(M(q'))} \right|, 1 \right)$$	Faithfulness of circuit to full-model response
GM Filter Consistency (Ahmed et al., 2023)	$$q(x) = (x - \bar{\mu})^\top \bar{\Sigma}^{-1}(x - \bar{\mu})$$; aggregated & compared to reference distribution	Fidelity of uncertainty estimates
Large Deviations (Dort et al., 2023)	Cost as $\int_0^1 (f'(s))^{\alpha'} ds$ in M1 topology	Topology-robustness; normalized rare event costs

7. Future Directions and Open Questions

Given NDF's demonstrated impact, several avenues are prominent:

• Further unification across disciplines, possibly with generalized formulations bridging discrete, continuous, and hybrid phenomena.
• Extension to domains such as causal inference, robust control, and reinforcement learning for model fidelity assessment.
• Theoretical analysis of normalization schemes to ensure invariance or stability under varying dataset characteristics or task definitions.

A plausible implication is that NDF-based metrics will become standard in evaluating the faithfulness, uncertainty, or risk modeling capabilities of next-generation AI and statistical systems.