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DirEAG: Dirichlet Evidence Aggregation for Calibrating Verbalized Confidence in Mathematical Reasoning
Haorui Xu, Yuzhou Zhu, Liyuan Gao
TL;DR
Black-box verbalized confidence is useful for mathematical reasoning but difficult to calibrate because repeated confidence-steered reports can have shifting scales. DirEAG converts each report into calibrated Dirichlet evidence over candidate answers plus a null state, and experiments show better calibration than direct or heuristic aggregation in many settings while retaining competitive answer selection.
Problem
Verbalized confidence and other black-box uncertainty signals do not reliably provide calibrated answer probabilities when confidence scales shift across prompts, models, and datasets.
Method
DirEAG calibrates each elicited confidence report as soft evidence over generated candidate answers and an explicit null state, then applies final binary calibration.
Results
Across GSM8K, SVAMP, and GSM-Hard, DirEAG often improves calibration over direct confidence averaging and heuristic steering aggregation while remaining competitive in answer selection.
Takeaways & Limitations
The results support treating verbalized confidence as informative but scale-dependent evidence rather than as a finished probability estimate.
Takeaways & Limitations
The study is limited to numerical mathematical reasoning with exact numeric matching and moderate-scale open-weight models, excluding open-ended, proof-oriented, multimodal, and semantically judged tasks.
Abstract
from arXiv · showhide
Reliable confidence estimation is essential for using large language models in mathematical reasoning, but black-box verbalized confidence is difficult to calibrate. When the same problem is queried under multiple confidence-steering prompts, the resulting answer-confidence observations contain useful uncertainty information, yet their scales may shift across steering levels, models, and datasets. Existing black-box uncertainty methods often rely on answer agreement, sample consistency, or entropy, which describe output variation but do not model the numerical meaning of self-reported confidence. Conversely, direct averaging or heuristic aggregation of elicited confidence cannot learn prompt- and task-dependent bias. We propose DirEAG, a Dirichlet Evidence Aggregation method that converts each elicited answer-confidence observation into calibrated soft evidence over generated candidate answers and an additional null state, allowing the model to represent cases where none of the candidates is correct. Experiments on GSM8K, SVAMP, and GSM-Hard with Qwen, Mistral, and Gemma models show that, compared with direct confidence averaging and heuristic confidence-steering aggregation, DirEAG often achieves better calibration while maintaining competitive answer selection. Ablations further reveal that evidence aggregation and final binary calibration address distinct parts of the calibration problem.
1 Introduction
DirEAG frames multi-prompt verbalized-confidence aggregation as a calibration problem: self-reports provide useful evidence, but their scales can shift across prompts and settings. It converts these reports into Dirichlet evidence over candidate answers and a null state, improving calibration in most evaluated settings while preserving competitive answer selection.
- Motivation: Existing uncertainty signals describe output variation, but do not directly model the numerical meaning of self-reported confidence.The paper contrasts answer agreement, sample consistency, and entropy with confidence reports whose scales may be biased.
- Method: DirEAG converts each confidence report into calibrated soft evidence over generated candidate answers and an explicit null state.The null state represents cases where all generated candidates are wrong, while the Dirichlet posterior combines observations before final binary calibration.
- Results: Across GSM8K, SVAMP, and GSM-Hard, DirEAG improves calibration over direct averaging and heuristic steering aggregation in most settings while remaining competitive in answer selection.Experiments use Qwen, Mistral, and Gemma models and compare against several confidence and uncertainty baselines.
- Motivation: Multi-prompt verbalized confidence is a statistical calibration problem because biased self-reports must be mapped to answer-level probabilities.Confidence scales may shift across prompts, models, and datasets, so raw self-reports should not be treated directly as probabilities.
- Method: Confidence evidence and final binary calibration play complementary roles: evidence affects candidate scoring and selection, while binary calibration corrects the selected-answer probability scale.The ablation separates these effects rather than treating calibration as a single operation.
2 Related Work
Prior black-box uncertainty work uses verbalized confidence, confidence steering, agreement, consistency, entropy, distractors, or fine-tuning. DirEAG focuses specifically on aggregating repeated answer-confidence observations after elicitation, preserving both candidate identity and reported confidence.
- Confidence calibration: Prior work studies confidence elicitation, probability baselines, and black-box aggregation, including verbalized-confidence overconfidence and multi-prompt steering.SteerConf queries the same model under multiple confidence prompts and aggregates the resulting observations.
- Related calibration methods: DINCO normalizes verbalized confidence against self-generated distractors, whereas ConfTuner fine-tunes models with a tokenized Brier-score objective.These approaches are complementary to DirEAG because they add distractor-based validation or modify the model rather than focusing on post-hoc aggregation.
- DirEAG’s scope: The paper focuses on aggregation after answer-confidence pairs are obtained, making repeated and conflicting observations part of the calibration problem.This separates the aggregation question from how confidence is initially elicited.
- Multi-observation uncertainty: Agreement-based methods estimate uncertainty from sampled-output distributions, agreement, or semantic diversity among multiple generations.Self-consistency, sample consistency, SelfCheckGPT, and semantic entropy represent this family of black-box signals.
- DirEAG’s distinction: Unlike agreement-frequency methods, DirEAG uses the joint structure of each confidence-steered observation: a candidate answer paired with a reported confidence score.For numerical reasoning, this yields a finite candidate set that can be modeled directly.
3 Method
DirEAG converts multiple confidence-steered answer reports into calibrated evidence over generated candidates and a null state, then calibrates selected-answer support as correctness probability. Its design addresses prompt-dependent confidence-scale bias while separating evidence aggregation from final binary calibration.
- Confidence elicitation: Five confidence-steering prompts produce answer-confidence observations whose scales may differ across steering levels.The method uses these observations as inputs rather than assuming their raw confidence values are directly comparable.
- Confidence elicitation: A monotonic Platt-style transform calibrates each raw confidence score with a shared positive slope and steering-level-specific bias.Clipping to (0, 1) precedes the logit-sigmoid transform because the logit is undefined at 0 and 1.
- Augmented candidate space: DirEAG aggregates calibrated soft evidence over a finite candidate-answer set and an explicit null state using a Dirichlet model.The null state represents cases where the gold answer is absent from the generated candidates.
- Augmented candidate space: Candidate evidence accumulates according to calibrated confidence and learned steering-level reliability weights, while low confidence contributes to null-state uncertainty.The model uses a variable candidate state space and places the null state on the same categorical simplex.
- Learning and binary calibration: The final answer is selected by maximum posterior probability, and its selected posterior mass is separately mapped to empirical correctness probability.The second Platt-style calibration corrects the binary probability scale required by ECE and Brier score without changing evidence aggregation.
- Learning and binary calibration: DirEAG trains its parameters by categorical negative log-likelihood with parameters shared within each calibration setting.Nonnegative parameters use positive parameterizations such as softplus.
4 Experiments
Experiments evaluate DirEAG across models, datasets, baselines, calibration metrics, and instance-level diagnostics. Results indicate that learned aggregation improves calibration while preserving competitive answer selection, with evidence aggregation and final binary calibration contributing distinct effects.
- Experimental Setup: Experiments span three models, three mathematical reasoning datasets, and metrics covering answer selection, probabilistic calibration, and confidence ranking.The models are Qwen2.5-7B-Instruct, Mistral-7B-Instruct-v0.3, and Gemma-2-9B-it; datasets are GSM8K, SVAMP, and GSM-Hard.
- Experimental Setup: DirEAG, mean confidence, and SteerConf use five answer-confidence observations, while self-consistency and answer-entropy use five vanilla samples.Top-K directly elicits five candidates at once, providing a single-response baseline.
- Main Results: Across models and datasets, agreement-based baselines remain strong for answer selection and ranking, but directly using agreement or reported scores does not consistently yield calibrated probabilities.Top-K is occasionally competitive in answer accuracy but can remain highly confident when wrong.
- Main Results: DirEAG achieves lower ECE in most settings than direct confidence averaging and SteerConf-style aggregation, while often improving answer selection.The result indicates that confidence-steered reports benefit from learned statistical aggregation.
- Main Results: Confidence quality has separate ranking and scale dimensions: AUROC and PR-N assess failure prioritization, whereas ECE and Brier score assess numerical calibration.The comparison therefore evaluates both probabilistic calibration and prioritization of likely errors.
- Internal Ablation: Adding confidence evidence can alter candidate scoring and selection but leaves raw posterior mass poorly calibrated; final binary calibration corrects the selected-answer probability scale without changing selection.Accuracy changes relative to count-only variants come from evidence aggregation, while ECE and Brier improvements mainly reflect final probability-scale correction.
- Diagnostic Analysis of Learned Uncertainty: DirEAG lowers Brier score than the constant diagnostic in all nine model–dataset settings, with absolute reductions of 0.021 to 0.086.It also achieves AUROC gains of 0.229 to 0.362 over the constant predictor, indicating instance-level uncertainty information.
- Diagnostic Analysis of Learned Uncertainty: The null-state probability is consistently higher when the gold answer is absent from generated candidates than when the final prediction is correct.This pattern suggests that the null state responds to candidate sets lacking the correct solution.
5 Conclusion
DirEAG treats verbalized confidence as informative but scale-dependent evidence rather than a finished probability estimate. Across three mathematical reasoning benchmarks, it improves calibration while remaining competitive in answer selection, with evidence aggregation and final calibration serving complementary roles.
- Conclusion: Across GSM8K, SVAMP, and GSM-Hard, DirEAG improves calibration and remains competitive in answer selection against direct and heuristic confidence baselines.The conclusion covers direct verbalized-confidence and heuristic confidence-steering comparisons.
- Conclusion: The ablation indicates that reported-confidence evidence and final calibration of the selected-answer score play complementary roles.The conclusion distinguishes the two components rather than treating them as interchangeable.
- Limitations: The study is limited to numerical mathematical reasoning with exact numeric answer matching and does not cover open-ended, dialogue, proof-oriented, or multimodal tasks.Extending DirEAG to those settings would require task-specific verification and candidate-state definitions.
Experimental Alignment and Ablation Variants
The ablation holds the five answer-confidence observations fixed while varying aggregation and calibration components. It also defines final binary calibration as a distinct variant-level design choice.
- Experimental alignment: Each black-box method uses five observations per problem, with steering prompts for DirEAG, mean confidence, and SteerConf, and vanilla samples for sample-based baselines.Top-K instead elicits five candidate answers in one response.
- Experimental alignment: For Mistral, the Top-K ranked candidates can conflict with the subsequent reasoning trace, including cases where the explanation derives the correct answer but the top listed candidate differs.This identifies a specific alignment issue in the direct multi-candidate baseline.
- Ablation setup: Each ablation variant reuses five answer-confidence observations and changes only aggregation and calibration.For level ℓ, (aℓ, qℓ) denotes the answer and confidence, while wℓ is the learned level weight.
- Ablation setup: The ablation compares variants that differ in whether they apply final binary calibration, denoted Cal(·).This isolates aggregation and calibration choices within a common setup.