How Independent are Large Language Models? A Statistical Framework for Auditing Behavioral Entanglement and Reweighting Verifier Ensembles

Chenchen Kuai, Jiwan Jiang, Zihao Zhu, Hao Wang, Keshu Wu, Zihao Li, Yunlong Zhang, Chenxi Liu

#41 of 2292 · Artificial Intelligence
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Tournament Score
1571±21
10501800
75%
Win Rate
82
Wins
27
Losses
109
Matches
Rating
6.2/ 10
Significance
Rigor
Novelty
Clarity

Abstract

The rapid growth of the large language model (LLM) ecosystem raises a critical question: are seemingly diverse models truly independent? Shared pretraining data, distillation, and alignment pipelines can induce hidden behavioral dependencies, latent entanglement, that undermine multi-model systems such as LLM-as-a-judge pipelines and ensemble verification, which implicitly assume independent signals. In practice, this manifests as correlated reasoning patterns and synchronized failures, where apparent agreement reflects shared error modes rather than independent validation. To address this, we develop a statistical framework for auditing behavioral entanglement among black-box LLMs. Our approach introduces a multi-resolution hierarchy that characterizes the joint failure manifold through two information-theoretic metrics: (i) a Difficulty-Weighted Behavioral Entanglement Index, which amplifies synchronized failures on easy tasks, and (ii) a Cumulative Information Gain (CIG) metric, which captures directional alignment in erroneous responses. Through extensive experiments on 18 LLMs from six model families, we identify widespread behavioral entanglement and analyze its impact on LLM-as-a-judge evaluation. We find that CIG exhibits a statistically significant association with degradation in judge precision, with Spearman coefficient of 0.64 (p < 0.001) for GPT-4o-mini and 0.71 (p < 0.01) for Llama3-based judges, indicating that stronger dependency corresponds to increased over-endorsement bias. Finally, we demonstrate a practical use case of entanglement through de-entangled verifier ensemble reweighting. By adjusting model contributions based on inferred independence, the proposed method mitigates correlated bias and improves verification performance, achieving up to a 4.5% accuracy gain over majority voting.

AI Impact Assessments

(3 models)

Scientific Impact Assessment

Core Contribution

This paper tackles a genuinely important and underexplored problem: quantifying the hidden behavioral dependencies ("latent entanglement") among LLMs that arise from shared training data, distillation, and alignment pipelines. The core novelty is a multi-resolution statistical framework comprising two information-theoretic metrics: (1) a Difficulty-Weighted Behavioral Entanglement Index (BEI) that captures synchronized failures weighted by task easiness, and (2) a Cumulative Information Gain (CIG) metric that captures directional alignment in erroneous responses (i.e., whether models select the same wrong distractor). The paper further demonstrates that these metrics predict judge bias in LLM-as-a-judge settings and proposes a de-entangled verifier ensemble reweighting strategy that improves over majority voting.

The key intellectual insight—that errors are more informative than correct answers for detecting dependence, since correct answers naturally converge while errors occupy a large hypothesis space—is sound and well-motivated. The hierarchical decomposition from binary failure co-occurrence to directional error alignment is a logical and elegant progression.

Methodological Rigor

Strengths: The statistical formulation is principled. The conditional independence null hypothesis (conditioning on task difficulty) is well-justified and draws appropriately from item response theory. The sign-flip randomization test for BEI and the Monte Carlo null distribution for CIG are appropriate nonparametric significance testing procedures. The logistic regression calibration for difficulty response functions is validated with AUC scores.

Concerns: Several methodological aspects raise questions:

1. Sample size and generalizability: The experiments use only 1,000 questions from MMLU-Pro, split into two subsets. This is a relatively small evaluation base for claims about "widespread behavioral entanglement." The restriction to a single benchmark (MCQ format) limits generalizability—entanglement patterns may differ substantially on open-ended generation, coding, or reasoning tasks.

2. CIG statistical significance inconsistencies: Table 3 reveals that several top CIG pairs have non-significant p-values (e.g., Qwen1.5-14B-Chat/Qwen1.5-72B-Chat at p=0.3975; Llama-2-70b-hf/Llama-3-70B at p=0.7131). These are presented alongside significant pairs without adequate discussion of why high CIG values can be statistically non-significant, which undermines confidence in CIG as a reliable metric.

3. Verifier ensemble evaluation: The de-entangled reweighting experiment uses only three judge models, which is a very small ensemble. The reported 4.5% accuracy gain, while notable, is demonstrated in a single experimental configuration without ablation over different ensemble sizes, hyperparameter sensitivity (κ, η₁, η₂, λ₁), or alternative benchmarks.

4. Causal claims vs. correlations: The Spearman correlations between CIG and judge bias (0.64, 0.71) are presented as evidence that entanglement *causes* over-endorsement bias, but the design only establishes association. Alternative confounds (e.g., model capability similarity) are not rigorously controlled for.

Potential Impact

The paper addresses a real and growing concern in the LLM ecosystem. As multi-model systems become standard—for evaluation, verification, safety, and red-teaming—understanding whether models provide truly independent signals is critical. The practical implications include:

  • LLM-as-a-judge pipelines: The finding that entanglement correlates with judge over-endorsement bias is directly actionable for evaluation infrastructure design.
  • Safety and verification: Redundancy-based safety systems that assume independence may have significantly lower effective coverage than expected.
  • Model selection: The entanglement graph (Figure 2) provides a practical tool for selecting maximally independent model ensembles.
  • The de-entangled reweighting strategy, while preliminary, demonstrates a clear path from diagnosis to mitigation.

    Timeliness & Relevance

    This work is highly timely. The LLM ecosystem is rapidly consolidating around a few foundation model families, and the concern about "model collapse" and homogenization is receiving increasing attention. The paper directly addresses a current bottleneck: the implicit independence assumption in multi-model evaluation and verification systems. The references to very recent models (GPT-5, Claude 4.6, Gemini 3.6) indicate the analysis covers the current frontier, though this also means the specific findings may have limited shelf life as models evolve.

    Strengths

    1. Novel problem formalization: The "failure manifold" perspective and the distinction between binary failure synchronization and directional error alignment constitute a meaningful conceptual contribution.

    2. Principled statistical approach: The conditional independence framework with proper null hypothesis testing goes beyond descriptive metrics.

    3. Breadth of models: 18 models from 6 families provides reasonable coverage of the ecosystem.

    4. Cross-family entanglement discovery: Finding entanglement between seemingly unrelated model families (e.g., DeepSeek-Gemini, Claude-GPT) is a surprising and important result.

    5. End-to-end pipeline: From metric definition to bias diagnosis to mitigation, the paper presents a complete workflow.

    Limitations

    1. Single benchmark, MCQ-only: The restriction to MMLU-Pro MCQ format fundamentally limits the scope of claims. The CIG metric specifically depends on discrete distractor choices.

    2. Small-scale verifier experiment: Three judges is insufficient to robustly evaluate the reweighting strategy.

    3. No comparison to existing dependence measures: Beyond Pearson correlation, there are established statistical dependence measures (mutual information, copulas, Hoeffding's D) that should be compared against.

    4. Reproducibility concerns: The use of proprietary API models (GPT-5, Claude 4.6) at specific snapshots makes reproduction difficult.

    5. Limited theoretical analysis: No formal guarantees on the statistical power of the proposed tests or the conditions under which BEI/CIG can distinguish true entanglement from other confounds (e.g., shared capability profiles).

    6. Missing ablations: The sensitivity of results to the number of tasks, choice of difficulty model, weighting schemes, and hyperparameters is not explored.

    Overall Assessment

    This paper makes a solid conceptual contribution to an important and timely problem. The statistical framework is sound in principle, and the empirical findings—particularly the cross-family entanglement patterns and the correlation with judge bias—are valuable for the community. However, the empirical evaluation is limited in scope (single benchmark, small ensemble), some statistical results are inconsistent (non-significant high-CIG pairs), and the practical mitigation strategy needs more thorough validation. The work would benefit substantially from extension to diverse task types, larger ensembles, and formal power analysis.

    Rating:6.2/ 10
    Significance 7.5Rigor 5.5Novelty 7Clarity 7

    Generated Apr 10, 2026

    Comparison History (109)

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