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Optimal Estimation of Watermark Proportions in Hybrid AI-Human Texts

Xiang Li, Garrett Wen, Weiqing He, Jiayuan Wu, Qi Long, Weijie J. Su

arXiv:2506.22343v2stat.MLcs.CLcs.LGstat.ME

TL;DR

The paper addresses how to estimate watermark proportions in texts mixing human and LLM-generated content, where binary detection and segment localization are insufficient. It formulates estimation through pivotal-statistic mixture models, proving non-identifiability for binary schemes and identifiability for continuous ones. Efficient estimators achieve minimax lower bounds and perform consistently well on synthetic and open-source-model mixtures.

  • Problem

    Estimating the overall watermark proportion in mixed-source text remains insufficiently explored despite its importance for authenticity and attribution.

    Source: §1 · p. 2, §3.2 · p. 6

  • Method

    The paper models watermark-preserving pivotal statistics as a mixture, establishes identifiability conditions, and develops efficient estimators with theoretical guarantees.

    Source: §1.1 · p. 2, §3.3 · p. 9

  • Results

    The proposed estimators achieve minimax lower bounds and consistently high accuracy on synthetic and open-source-model mixed-source texts.

    Source: §5.2 · p. 15, §6 · p. 17

  • Takeaways & Limitations

    Continuous pivotal statistics support meaningful watermark-proportion estimation, whereas binary green-red statistics can make the proportion fundamentally unidentifiable.

    Source: §6 · p. 17

    The estimation model assumes approximately position-homogeneous watermark retention and does not account for the importance or semantics of generated content.

    Source: §6 · p. 17

Abstract

from arXiv · show

Text watermarks in large language models (LLMs) are an increasingly important tool for detecting synthetic text and distinguishing human-written content from LLM-generated text. While most existing studies focus on determining whether entire texts are watermarked, many real-world scenarios involve mixed-source texts, which blend human-written and watermarked content. In this paper, we address the problem of optimally estimating the watermark proportion in mixed-source texts. We cast this problem as estimating the proportion parameter in a mixture model based on \emph{pivotal statistics}. First, we show that this parameter is not even identifiable in certain watermarking schemes, let alone consistently estimable. In stark contrast, for watermarking methods that employ continuous pivotal statistics for detection, we demonstrate that the proportion parameter is identifiable under mild conditions. We propose efficient estimators for this class of methods, which include several popular unbiased watermarks as examples, and derive minimax lower bounds for any measurable estimator based on pivotal statistics, showing that our estimators achieve these lower bounds. Through evaluations on both synthetic data and mixed-source text generated by open-source models, we demonstrate that our proposed estimators consistently achieve high estimation accuracy.

1 Introduction

The paper studies estimating watermarked-content proportions in mixed human–LLM texts, extending beyond binary detection and segment localization. It establishes identifiability limits, efficient estimators, minimax guarantees, and empirical validation.

1 Introduction

  • Mixed-source texts make a single proportion estimate more practical than precise segment localization for authenticity and threshold-based decisions.Existing localization methods can be statistically infeasible, while aggregate detection may remain possible even when individual signals cannot be located.Source: §1 · p. 2

1.1 Our Contributions

  • The paper formulates watermark-proportion estimation as a mixture-model problem based on pivotal statistics.Source: §1.1 · p. 2
  • The framework analyzes what to estimate, how to estimate it, attainable accuracy, and minimax lower bounds, with proposed estimators achieving theoretical optimality.The estimators are computationally efficient and achieve O(n) complexity using a logarithmic number of passes.Source: §1.1 · p. 3, §1.2 · p. 3

1.2 Related Work

  • Prior work mainly addresses binary detection or watermarked-segment localization, not systematic estimation of the overall watermark proportion.Existing localization methods include span search, dynamic programming, online learning, and change-point detection approaches.Source: §1.2 · p. 3

2 Preliminaries

The preliminaries define watermark verification through token-level pivotal statistics derived from pseudorandomness and explain the structural differences among three representative watermarking schemes.

2 Preliminaries

  • LLMs autoregressively sample tokens from unknown next-token prediction distributions determined by preceding context.Source: §2 · p. 3
  • Watermark verification reduces to testing dependence between generated tokens and reconstructible pseudorandom variables through pivotal statistics.Human-written tokens are independent of the pseudorandomness, whereas watermark decoding introduces dependence.Source: §2 · p. 4
  • The framework studies Gumbel-max, inverse transform, and green-red list watermarks, which differ in their decoders and pseudorandom-variable constructions.Once pivotal statistics are specified, each scheme also determines its null and alternative distributions.Source: §2 · p. 5

3 Proportion Estimation Methods

The paper estimates the proportion of watermark-preserving pivotal statistics under unknown, varying NTP distributions. It proves that binary statistics can make the proportion non-identifiable, while continuous statistics enable identification and efficient estimation under regularity conditions.

3.1 Target Estimand

  • The estimand ε is the marginal proportion of watermark-preserving pivotal statistics, not necessarily the fraction of modified tokens.The model assumes this retention rate is constant across positions and is most appropriate for approximately position-homogeneous editing.Source: §3.1 · pp. 5–6

3.2 Estimability

  • Identifiability means uniquely determining ε from observed pivotal statistics despite unknown and potentially varying NTP distributions.Without identifiability, multiple ε values can produce the same data distribution and undermine interpretation of the estimate.Source: §3.2 · p. 6
  • For green-red list watermarks, ε is non-identifiable because binary pivotal statistics provide insufficient information about the unknown mixture parameters.The result follows both in a simplified binary mixture and under the practical model with varying alternative probabilities.Source: §3.2 · p. 6
  • For continuous pivotal statistics satisfying Assumption 3.1, ε is identifiable, including for Gumbel-max and inverse transform watermarks.The inverse-transform result uses exact finite-vocabulary CDFs without requiring the asymptotic condition used for a limiting approximation.Source: §3.2 · p. 7

3.3 Estimation Methods

  • The estimation methods use moment equations, empirical alternative CDFs, and weight functions to avoid estimating each high-dimensional NTP distribution directly.Directly estimating distributions over vocabularies such as OPT-1.3B’s 50,272-token vocabulary is infeasible at typical text lengths.Source: §3.3 · pp. 7–8
  • The initial small-threshold estimator is simple but statistically inefficient and heuristic because it uses few observations when δ is small.Source: §3.3 · p. 8
  • The variance bound is tight when NTP distributions are identical and deterministic, establishing a matching minimax lower bound in that regime.Source: §3.3 · p. 8
  • The optimal estimator uses an empirically estimated weight function inside a fixed-point operator, which generally has a unique solution when the density-ratio estimate is sufficiently accurate.The method balances information from the estimated alternative CDF with consistency for the unknown ε.Source: §3.3 · p. 9

4 Theoretical Analysis

The theoretical analysis characterizes the accuracy and limits of watermark-proportion estimators, showing when refinements attain optimal error and when non-identifiability causes irreducible bias.

4.1 Accuracy for Initial Attempts

  • Under Assumption 3.1, the initial estimator’s right-hand error bound vanishes when δ_n → 0 and nF_0(δ_n) → ∞.Its MAE combines a noise term from sampling with bias caused by omitting the unknown alternative-distribution term.Source: §4.1 · p. 10
  • Adaptively tuning δ can reduce the initial estimator’s bias and make it asymptotically unbiased under the stated conditions.The bias arises from neglecting the unknown averaged alternative CDF and can be mitigated as δ becomes sufficiently small.Source: §4.1 · p. 10

4.2 Accuracy for Refinements

  • The refined estimators achieve analyzed MAE bounds when the density-ratio estimate is sufficiently accurate, while the optimal-weight estimator achieves smaller MAE and greater stability.Analyzing the optimal estimator additionally requires moment conditions for the fixed-point solutions, and its idealized derivation assumes the averaged alternative CDF is known.Source: §4.2 · pp. 10–11

4.3 Minimax Lower Bounds

  • The minimax lower bounds are tight: the refined estimator matches the optimal rate up to the error from estimating the averaged alternative CDF, whereas the initial estimator needs its correction term to attain that rate.The bounds depend on aggregate alternative behavior rather than the order-specific sequence of token distributions, which is typically unavailable in verification settings.Source: §4.3 · p. 11

4.4 Inherent Bias Caused by Non-identifiability

  • For the green-red list watermark, the bias does not vanish as δ → 0 because the scheme violates Assumption 3.1 and the proportion is non-identifiable.The binary signals provide negligible information about the unobservable averaged alternative parameter, so even MLE can remain biased with large samples and small regularization.Source: §4.4 · p. 12

5 Experiments

Experiments evaluate the estimators on synthetic mixtures and open-source LLM text, showing that OPT is generally the most accurate and stable method while non-identifiability causes systematic failure for binary watermark settings.

5.1 Simulation Study

  • As ε increases, Gumbel-max mixture CDFs depart further from the uniform null distribution, with ε=1 yielding the greatest deviation.The CDFs are shown for ∆=0.1; additional watermark settings appear in the appendix.Source: §5.1 · pp. 13–14, Figure 1 · p. 13
  • OPT consistently outperforms INI and IND across ground-truth ε values when ∆=0.1.The compared estimators are INI, refined IND, and refined OPT with the optimal weight function.Source: Figure 1 · p. 13, §5.1 · p. 14
  • OPT achieves the lowest averaged MAEs and smallest standard deviations across watermark strengths; for Gumbel-max with ∆=0.1, it records 11 × 10−4 versus 210 × 10−4 for INI and 23×10−4 for IND.The optimal weight function improves estimation by minimizing estimator variance.Source: §5.1 · p. 14, Table 2 · p. 14
  • Binary green-red list watermarks produce non-identifiable proportions, with INI often estimating zero and IND or OPT estimating values near one.The failure follows because inaccessible NTP distributions and binary observations provide insufficient information about ε.Source: §5.1 · p. 14
  • Regularized MLE estimates remain biased for i.i.d. mixture binary data even with large n and small λ, confirming that non-identifiability is inherent to the model.The empirical estimates track the theoretical non-identifiability limit rather than converging to the true ε.Source: Figure 1 · p. 13, §5.1 · pp. 14–15

5.2 Open-source Model Experiments

  • The open-source evaluation mixes watermarking probabilistically at token level, uses repeated-context masking, and compares against WPL’s token-level localization baseline.Repeated-context masking lowers realized watermark proportions below nominal ε, while WPL estimates the global proportion by averaging token labels.Source: §5.2 · p. 15
  • On OPT-1.3B and C4, OPT has the smallest MAE and variation in nearly all open-source experiments, while WPL performs worse than every proposed estimator.Figure 2 varies true proportions and temperature, and Table 3 summarizes results across models, temperatures, and watermarks.Source: §5.2 · p. 15, Figure 2 · p. 16, Table 3 · p. 16
  • Under random substitution, deletion, and insertion, OPT usually retains the lowest MAE and variance, although INI or IND occasionally performs better.Post-processing modifies pivotal statistics through token context, and the resulting C4 and arXiv evaluations show similar findings.Source: §5.2 · pp. 15–16, Table 4 · p. 17

6 Discussion

The paper establishes when watermark proportions can be estimated in mixed-source texts and identifies important boundaries for interpreting those estimates.

6 Discussion

  • Binary pivotal statistics make the proportion non-identifiable, whereas continuous pivotal statistics permit identifiability and minimax-optimal estimation.The proposed estimators perform strongly on synthetic and LLM-generated data.Source: §6 · p. 17
  • The estimand measures AI-generated extent but not the importance of individual content, so semantic information could support more nuanced estimates.The paper also identifies token-level modeling and unexploited sequential structure as scope boundaries.Source: §6 · p. 17
  • Theoretical guarantees require sufficiently small density-ratio estimation error, which can be difficult to achieve under distributional shifts.More robust procedures that relax this requirement remain an open question.Source: §6 · p. 18

Appendices

The appendix proof setup defines predictive distributions for pivotal statistics generated from random, autoregressive next-token distributions.

Appendices

  • The proof treats the surrogate empirical CDF as deterministic because it is computed before estimation.This establishes the fixed reference quantity used in the analysis.Source: A Proofs of Theoretical Analysis · p. 22
  • For autoregressively generated tokens, pivotal statistics are analyzed through conditional CDFs given the past filtration.The corresponding predictive quantities depend on random and unknown next-token distributions.Source: A Proofs of Theoretical Analysis · p. 22

A.1 Proof of Results in Section 3.2

The appendix proves identifiability and supporting efficiency results under regularity conditions, while characterizing how the watermark constructions satisfy those conditions.

A.1 Proof of Results in Section 3.2

  • Multiple pairs of mixture proportion and alternative distribution can produce the same Bernoulli law, proving non-identifiability for the binary case.This is the proof mechanism behind the green-red list watermark limitation.Source: §A.1 · p. 22
  • The proportion is identifiable because a boundary ratio is determined by the observed marginal distribution when the null CDF is known.The argument uses the limiting behavior of the alternative-to-null CDF ratio near zero.Source: §A.1 · pp. 22–23
  • Gumbel-max and inverse-transform watermarks satisfy the regularity conditions through absolute continuity, bounded density ratios, and non-singular next-token distributions.For inverse transform, the alternative statistic is bounded away from zero near the origin, while the null CDF has linear behavior there.Source: §A.1 · p. 23
  • The inverse-transform verification does not require an asymptotic condition on P(2).The proof establishes the needed uniform bounds over the relevant distributions.Source: §A.1 · pp. 24–25

A.2 Proof of Lemma 3.4

  • The optimal weighting function is proportional to vopt(ε, x), with equality characterized by the Cauchy-Schwarz step.The appendix derives the optimizer after centering the candidate function under the mixture distribution.Source: §A.2 · p. 25

A.3 Proof of Inequality (6)

  • The variance inequality becomes equality only under restrictive conditions, including matching conditional distributions and, ultimately, an i.i.d. pivotal-statistic sequence.The proof uses conditional variance, Jensen’s inequality, and the law of total variance.Source: §A.3 · p. 26

A.4 Analysis for the Initial Estimator

The initial estimator’s error decomposes into stochastic and bias terms, and adaptive threshold selection improves its rate without reaching minimax optimality.

A.4 Analysis for the Initial Estimator

  • Theorem 4.1 gives the initial estimator’s MAE as order σ_n/√n plus a bias term.The stochastic component comes from randomness in the collected data.Source: §A.4 · p. 27
  • The estimator’s variance can be smaller than the minimax variance up to constants, but its bias generally increases overall MAE.The density-ratio contribution makes the initial estimator’s error larger.Source: §A.4 · p. 27
  • When the alternative and null CDFs behave as x^p and x near zero with p > 1, the optimal threshold depends on n more slowly than 1/√n.This slower dependence explains why the initial estimator cannot achieve minimax optimality.Source: §A.4 · p. 28

A.5 Analysis for the Refined Estimator

This section establishes refined-estimator guarantees and develops the supporting contraction analysis for the optimal estimator.

A.5 Analysis for the Refined Estimator

  • The refined estimator’s theorem provides a finite-sample accuracy guarantee under the stated mixture and moment conditions.The supplied theorem and corollary passages state the result but omit enough notation to reproduce its exact bound.Source: §A.5 · p. 28
  • The estimator analysis decomposes the empirical mixture CDF through conditional probabilities and Jensen’s inequality.The decomposition uses p_t(ε) and the relation defining the projected empirical distribution.Source: §A.5 · pp. 28–29

A.6.1 Proof of Lemma 3.5

  • The supporting lemma defines a projected operator on [εmin, 1 − εmin] and analyzes its contraction using total variation separation and density-ratio conditions.The projection prevents numerical issues at the boundary, while Lemma A.1 supplies the high-probability control needed for Lemma 3.5.Source: §A.6.1 · p. 29
  • Corollary 5 gives high-probability control of the operator’s bound, with the form depending on whether an additional moment condition holds.The proof uses the projection operator and bounds involving C, log n, and n.Source: §A.6.1 · p. 30

A.6.2 Proof of Theorem 4.2

This section develops the theoretical and empirical analyses supporting the proposed estimators, including error bounds, concentration results, minimax lower bounds, regularized maximum-likelihood estimation, and numerical procedures. It also describes the NTP-distribution construction and empirical CDF behavior used in evaluation, along with additional estimator results and limitations for green-red list watermarks.

A.6.2 Proof of Theorem 4.2

  • Theorem A.2 bounds E|bεopt−ε| by variance, bias, boundary, and higher-order terms under total-variation and boundedness assumptions.The proof analyzes the optimal estimator through a fixed-point equation and an L-Lipschitz operator.Source: §A.6.2 · p. 31, §A.6.4 · p. 38
  • With ∆g→0, the bias satisfies B≤Cg∆g^2DTV(F0,bFP)=og(1), while the variance and auxiliary terms yield the stated simplified MAE.If EF0[g]^-2 is unavailable, εmin=log n/n^1/4 requires the stronger condition n^1/4·∆g=o(1).Source: §A.6.2 · pp. 31–32

A.6.3 Proof of Lemma A.1

  • Lemma A.1 controls the predictive mean and uniform error of the optimal weighting function using total-variation bounds, concentration, and separate moment-condition cases.The proof bounds the estimator’s component terms uniformly over ε∈[εmin,1−εmin].Source: §A.6.3 · pp. 33–36

A.6.4 Proof of Theorem A.2

  • Theorem A.2 follows by comparing empirical and population fixed points, applying the operator’s Lipschitz continuity, and bounding predictive fluctuation through Lemma A.1.The boundary projection contributes εmin·1{ε∉[εmin,1−εmin]} when the true proportion lies outside the interior interval.Source: §A.6.4 · pp. 38–39

A.7 Proof of Lemma A.2

  • Lemma A.2 handles dependence among pivotal statistics with martingale differences, while η-net discretization gives uniform control over ε through Lipschitz bounds and a union bound.The parameter interval is discretized with mesh η=εmin^3/(2n), reducing uniform deviation to finitely many pointwise bounds.Source: §A.7 · p. 39, §A.7.1 · pp. 39–40

A.7.1 η-net Discretization

  • The η-net reduction applies for both k = 1 and k = 2, after which pointwise tail probabilities are bounded separately for each function class.Source: §A.7.1 · p. 40

A.7.2 Without Moment Conditions

  • Without moment assumptions, Azuma–Hoeffding controls the martingale terms for H1 and H2, yielding concentration bounds with explicit 1/ε2 dependence.Source: §A.7.2 · p. 40

A.7.3 With Moment Conditions

  • With a predictable quadratic-variation bound, Freedman’s inequality gives sharper concentration; EF0[bg−2] < ∞ and bounded likelihood ratios ensure the required deterministic bound.The predictable quadratic variation is deterministically bounded under these conditions.Source: §A.7.3 · pp. 41–42

A.8 Proof of Theorem 4.5

  • Theorem 4.5’s lower bound restricts to an i.i.d. subclass and applies Le Cam’s two-point method to indicator-based and general measurable estimators.For general estimators, the proof uses total-variation control through KL and chi-square divergences, including data processing.Source: §A.8 · pp. 42–45

A.9 Proof of Theorem 4.6

  • Theorem 4.6 shows that L2 regularization makes the MLE solution exist uniquely, subject to a valid mixture-domain constraint, and establishes its limiting behavior as λ → 0.The proof also bounds the fitted mixture and shows all subsequences converge to the same limits.Source: §A.9 · pp. 46–47

B.1 Further Details

  • The implementation approximates the optimal-weight integrals with finite samples, then solves the refined estimator’s bounded optimization problem using SciPy’s L-BFGS-B method.It uses εmin = 10−3 and initializes optimization by iterating the estimated operator 20 times from 0.9.Source: §B.1 · pp. 47–48
  • NTP distributions combine a Zipf-like dominant head with a uniform tail, while enforcing that the largest probability remains below 1 −∆.The construction uses |W| = 10^3, randomizes Zipf parameters and support size, and returns a randomly permuted distribution.Source: §B.1 · pp. 48–49
  • For green-red list watermarks, INI and IND are highly biased when NTP distributions are inaccessible, so projected estimators are used to keep MAEs within [0, 1].The projection preserves the applicable MAE analysis because projection is nonexpansive; the watermark parameters are (γ, δ) = (0.5, 2).Source: §B.1 · p. 48

B.2 Empirical CDFs of Mixture Distributions

  • The mixture CDF becomes progressively more right-skewed as ε increases, with the ε = 1 alternative stochastically larger than the ε = 0 null distribution.Larger ∆ increases the discrepancy between F0 and F̄P, thereby increasing mixture-CDF right-skewness.Source: §B.2 · p. 49

B.3 Additional Results on Estimation Accuracy

  • In Figures 5 and 6, OPT achieves the lowest MAE in most cases for Gumbel-max and inverse transform watermark estimation.The figures report the complete estimation-performance results for both watermark types.Source: §B.3 · p. 49

B.4 Additional Results on Inherent Bias

This section examines inherent bias through mixture CDFs and regularized maximum-likelihood estimates for two watermarking schemes. Across parameter settings, the estimates closely match the theoretical limiting proportion, supporting the predicted non-identifiability bias.

B.4 Additional Results on Inherent Bias

  • Figures 3 and 4 plot mixture CDFs for the Gumbel-max and inverse transform watermarks, respectively, using ∆ and ε as the dominance and true-proportion parameters.The supplied captions identify the plotted quantities and the two watermarking schemes.Source: Figure 3 · p. 50, Figure 4 · p. 51
  • For the Gumbel-max watermark, the alternative CDF is stochastically larger than the null, and increasing ε makes the mixture CDF progressively more right-skewed.The cited prose describes the corresponding mixture behavior illustrated by the CDF plots.Source: Figure 3 · p. 50
  • Figures 5 and 6 report MAEs for the Gumbel-max and inverse transform watermarks, selecting the best values over δ ∈ {10^-1, 10^-2, 10^-3}.The figures report MAEs for both INI bεini(δ) and IND bεrfn(δ).Source: Figure 5 · p. 52, Figure 6 · p. 53
  • Across all tested (γ, µ) combinations, L2-regularized MLEs closely align with the theoretical limit proportion, supporting inherent bias.Figure 7 compares λ = 10^-2 estimates at n ∈ {10^3, 10^4, 10^5} with the limit from Theorem 4.6.Source: §B.4 · pp. 49, 55, Figure 7 · p. 54

C.1 Details on Mixture Data Setting

The mixture-data experiments generate pivotal statistics from watermarked and null samples, then compare estimators across true proportions and model settings. The setup accounts for masking effects that can reduce the effective watermark proportion.

C.1 Details on Mixture Data Setting

  • The synthetic generation uses a context window of m = 4 and applies watermarking independently with probability ε when constructing the target mixture.Repeated-context masking can prevent some intended watermark applications, reducing the effective proportion.Source: §C.1 · p. 55
  • Prompted C4 documents contain at least 50 tokens, with 20 buffer tokens generated to reduce the need for model-dependent padding.The verifier is ensured at least n tokens despite possible retokenization differences.Source: §C.1 · p. 55
  • Mixed-source datasets combine approximately ε-proportion watermarked samples with (1 − ε)-proportion null samples, using pivotal statistics rather than tokens for estimator evaluation.The experiments aggregate watermarked pivotal statistics into a fully watermarked dataset for estimating auxiliary quantities used by refined estimators.Source: §C.1 · p. 55
  • For a fair comparison, WPL tunes its threshold over predefined sets, using {1.3, 1.5, 1.7} for Gumbel-max and {0.73, 0.75, 0.77} for inverse-transform watermark, and reports the lowest-MAE result.Source: §C.1 · pp. 55–56
  • Figure 8 evaluates estimator mean absolute errors across true proportions ε and temperature parameters for OPT-13B on C4.The figure provides the experimental grid for assessing estimation accuracy under varying mixture proportions and temperatures.Source: Figure 8 · p. 56

C.2 Additional Results on Estimation Accuracy

Across additional open-source-model experiments, OPT generally provides the strongest estimation accuracy, including under post-processing modifications. The accompanying setup defines the ground truth through pivotal statistics associated with the alternative distribution.

C.2 Additional Results on Estimation Accuracy

  • OPT achieves the lowest MAE in most C4 and arXiv scenarios, while WPL performs poorly when the true proportion is around 0.5.Figures 8–10 cover OPT-13B, LLaMA3.1-8B, and three models on arXiv.Source: §C.2 · p. 56, Figure 9 · p. 57, Figure 10 · p. 58

C.3 Details on Post-processing Modification

  • The post-processing experiments evaluate three LLMs on C4 and arXiv after random token substitutions, deletions, or insertions at specified fractions.Prompts use the last 50 document tokens, and each continuation contains 500 generated tokens at temperature 1.Source: §C.3 · p. 59
  • The post-processing ground truth is the fraction of pivotal statistics following the alternative distribution, treating statistics from segments appearing in the modified text as watermarked.This defines the true ε through the condition Y_t|P_t ∼ µ_1,P_t.Source: §C.3 · p. 59

C.4 Additional Results on Post-processing Modification

  • Under post-processing modifications, OPT again achieves the lowest MAE and variance in most arXiv cases, although INI or IND occasionally performs better.Table 5 averages MAEs over 11 ground-truth ε values and reports standard deviations in parentheses, with values in units of 10−3.Source: §C.4 · p. 59, Table 5 · p. 59
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