Technical Summary: Disentangling Ambiguity from Instability in Large Language Models

Problem Statement

Deploying Large Language Models (LLMs) for clinical Text-to-SQL applications faces a critical barrier: the risk of generating syntactically correct but semantically erroneous queries. A primary challenge is latent ambiguity, where input queries contain implicit uncertainties (e.g., "patients over 18" referring to age at diagnosis vs. current age) that significantly alter results but are not evident to the user.

Current uncertainty quantification methods, such as Kernel Language Entropy (KLE), provide a single score to measure output diversity. However, this approach conflates two fundamentally distinct sources of uncertainty:

1. Input Ambiguity (Aleatoric-like): Diversity arising from the inherent ambiguity of the user's query, which legitimately yields multiple valid interpretations.
2. Model Instability (Epistemic-like): Diversity arising from the model's internal inconsistency or lack of knowledge when processing a single, clear interpretation.

This conflation hinders effective system deployment. High ambiguity should trigger a clarification dialogue with the user, whereas high instability should flag the query for human review. Existing methods cannot distinguish between these regimes, preventing targeted interventions.

Methodology: The CLUES Framework

The authors propose CLUES (Conditional Language Uncertainty via Entropy and Schur), a framework that models Text-to-SQL as a two-stage generative process to decompose semantic uncertainty.

1. Two-Stage Generative Process

The system is formalized as:

• Interpretation Stage: The system generates $N$ distinct semantic interpretations ($I = \{I_1, ..., I_N\}$) representing plausible readings of the original query.
• Generation Stage: For each interpretation $I_n$, the system generates $M$ candidate SQL queries and corresponding answers ($R_n = \{R_{n,1}, ..., R_{n,M}\}$).
  This structure creates a bipartite relationship where diversity across interpretations represents ambiguity, and diversity within a single interpretation represents instability.

2. Bipartite Semantic Graph Construction

A similarity function $k(\cdot, \cdot)$ is used to construct a weighted adjacency matrix $W$ for the entire system, structured as a bipartite graph:
$$W = \begin{pmatrix} W_{II} & W_{IR} \\ W_{RI} & W_{RR} \end{pmatrix}$$

• $W_{II}$: Encodes semantic similarity among interpretations.
• $W_{RR}$: Encodes semantic similarity among all generated answers.
• $W_{IR}/W_{RI}$: Binary assignment matrices linking answers to their parent interpretations.

A heat diffusion kernel ($K_\tau = e^{-\tau L}$) is applied to smooth these similarities, where $\tau$ controls the granularity of clustering.

3. Decomposition via Schur Complement

The core innovation is the decomposition of total uncertainty into an Ambiguity Score ($H_I$) and an Instability Score ($H_{R|I}$).

• Ambiguity ($H_I$): Computed as the entropy of the interpretation matrix $W_{II}$ using the KLE framework.
• Instability ($H_{R|I}$): Defined as the conditional entropy of results given interpretations. The authors reject the naive subtraction approach ($H_{R,I} - H_I$) because it violates the chain rule for von Neumann entropy on heat-kernel density matrices and frequently yields negative values.
• Schur Complement Solution: Instead of subtraction, the framework computes the conditional similarity structure directly:
  $$S = W_{RR} - W_{RI}(W_{II} + \epsilon I)^{-1}W_{IR}$$
  The matrix $S$ represents the residual similarity in answers that cannot be explained by the interpretation structure. The entropy of $S$ yields $H_{R|I}$. This approach guarantees non-negative scores and mathematically isolates the system's internal inconsistency.

4. Uncertainty Regimes

The framework defines four operational regimes based on thresholds for $H_I$ and $H_{R|I}$:

• Regime I (Confident): Low ambiguity, low instability $\rightarrow$ Auto-answer.
• Regime II (Ambiguity): High ambiguity, low instability $\rightarrow$ Trigger clarification dialogue.
• Regime III (Instability): Low ambiguity, high instability $\rightarrow$ Flag for human review.
• Regime IV (Compound): High ambiguity, high instability $\rightarrow$ Clarification + Review.

Key Contributions

1. CLUES Framework: A model-agnostic method to decompose semantic uncertainty into ambiguity and conditional instability using the Schur complement of a bipartite semantic graph.
2. Clinical Dataset & Procedure: An interpretation-generation procedure for epidemiological questions and a clinical Text-to-SQL benchmark featuring multiple valid interpretations for known queries.
3. Empirical Validation: Evidence demonstrating that $H_{R|I}$ improves failure prediction over state-of-the-art KLE ($H_R$) and enables effective error triage through regime-based routing.

Experimental Results

Open-Domain QA (AmbigQA & SituatedQA)

Using gold-standard interpretations, CLUES was evaluated across five frontier LLMs.

• Failure Prediction: The instability score $H_{R|I}$ consistently outperformed the baseline $H_R$ in predicting failures (AUROC improvements of +0.074 to +0.077 on pooled data).
• Regime Discrimination: Among queries with high output diversity (high $H_R$), those with low instability ($H_{R|I}$) had significantly higher accuracy than those with high instability. For example, in SituatedQA, accuracy dropped from 46.8% (Regime A) to 13.3% (Regime B), a statistically significant difference (odds ratio ~5.7).
• Naive Subtraction Failure: The naive subtraction approach ($\tilde{H}_{R|I}$) yielded negative values in 37–60% of cases and performed near chance, validating the necessity of the Schur complement.

Clinical Text-to-SQL (Known Interpretations)

Evaluated on a dataset derived from EHRSQL and EpiAskKB with 2,180 query-model instances.

• Performance: $H_{R|I}$ achieved a pooled AUROC of 0.762 compared to 0.600 for $H_R$ (p < $10^{-10}$).
• Robustness: The instability score remained stable across different heat kernel parameters ($\tau \in [2, 20]$), whereas the baseline $H_R$ degraded significantly at higher $\tau$.
• Temperature Sensitivity: Moderate sampling temperatures (T=1.0–1.5) yielded the strongest instability signals, suggesting higher diversity exposes latent model instability.

Deployment Setting (Generated Interpretations)

In a production-like setting where interpretations are generated on-the-fly (not given as gold labels):

• Accuracy: Models achieved high accuracy (84–100%), resulting in sparse errors (4.4% error rate).
• Regime Efficiency: While single-score AUROC was not significantly different between $H_R$ and $H_{R|I}$ in this sparse regime, the regime analysis proved valuable. Regime IV (High Ambiguity + High Instability) contained 51% of all errors while covering only 25% of queries.
• Routing: A strategy routing only Regime IV to human review would catch half of all failures by reviewing a quarter of queries, doubling the efficiency of random selection.

Significance and Claims

The paper claims that CLUES provides a diagnostic decomposition unavailable from single-score metrics. By distinguishing between input ambiguity and model instability, the framework enables targeted interventions:

• Clarification for high-ambiguity queries.
• Human Review for high-instability queries.

The authors emphasize that this distinction is operationally critical for safety-critical domains like clinical EHR access, where "plausible yet incorrect" results pose significant risks. The framework is designed to be model-agnostic, requiring only output sampling, and offers a principled mathematical approach to uncertainty that avoids the pitfalls of naive entropy subtraction.

The authors modestly note limitations, including the computational cost of multiple LLM calls per query and the dependency on the quality of generated interpretations. They also acknowledge that in high-accuracy deployment settings, the sparse error rate limits the discriminative power of fine-grained uncertainty signals, though regime-based routing still offers efficiency gains over random selection.