Improving Language Model Confidence Estimation with Linear Representations of Ambiguity

Core problem: language models estimate their own answer correctness confidence inaccurately; the authors want to use the model's internal representation of input ambiguity to improve confidence calibration.

• Setting and motivation: the authors argue that as LMs are deployed in high-stakes settings, accurately estimating the likelihood of success on a given input is critical for downstream decisions (e.g., requesting clarification, handing off to retrieval); existing work shows LMs are poor estimators of their own success probability.
• Core hypothesis: poor calibration partly stems from failure to detect aleatoric uncertainty (ambiguity inherent in the input, e.g., "bank" can mean a financial institution or a riverbank); the authors hypothesize that LMs internally encode at least part of the input ambiguity information in the form of linear representations. The authors note they focus only on aleatoric uncertainty, not epistemic uncertainty related to model knowledge and varying by model.
• Shortcomings of existing methods: per the paper, existing interpretability work has found representations related to answer truthfulness, but the existence of a representation does not mean the model can output calibrated confidence; recent work using representation interventions on behavior mostly targets refusal, sycophancy, and hallucination, not calibration.
• What the paper proposes: (1) use difference-in-means to extract an ambiguity direction on ambiguous/disambiguated minimal pairs and verify its causal effect; (2) propose DART (Direction-Aligned Regularization for Tuning), a LoRA fine-tuning method that restricts updates to within the ambiguity subspace, encoding the calibration improvement into the model parameters.

Confidence estimation and calibration

• Post-hoc calibration methods: histogram regression (Nobel, 1996), Platt scaling (Platt, 1999), isotonic regression (Barlow et al., 1972), beta calibration (Kull et al., 2017), etc., which recalibrate a single confidence score.
• Ways to obtain LM confidence: sequence joint probability (Desai and Durrett, 2020), direct questioning (linguistic calibration; Mielke et al., 2022; Tian et al., 2023; Band et al., 2024), using the model's internal states (Liu et al., 2024; Subramani et al., 2025, 2026; Kumaran et al., 2026).

Ambiguity in NLP

• Pavlick and Kwiatkowski (2019) show that NLI annotation disagreement reflects differences in plausible interpretations; Plank (2022) advocates directly modeling label disagreement; Baan et al. (2022) show that standard calibration metrics fail on inputs with multiple interpretations.
• Ambiguity benchmarks: open-domain question answering ambiguity (Min et al., 2020), scope ambiguity (Kamath et al., 2024), AMBIENT (Liu et al., 2023) and the ambiguity taxonomy (Li et al., 2024); the paper states that results on these evaluations are mixed.

Actionable interpretability

• Early "BERTology" localized representations of linguistic structure (Hewitt and Manning, 2019; Tenney et al., 2019, etc.); more recently it has been argued that interpretability should be evaluated by downstream usefulness (Nanda et al., 2025; Orgad et al., 2026).
• Work using representations to intervene on behavior mostly targets refusal (Arditi et al., 2024), sycophancy (Chen et al., 2025), hallucination (Prasad et al., 2026); the authors position this paper's difference as: targeting confidence calibration and directly improving calibration with a small number of directions.

Steering vectors and baselines

• The most common method for locating steering vectors is difference-in-means (Marks and Tegmark, 2024; Rimsky et al., 2024), which this paper follows.
• Experimental baselines: base model, LoRA without regularization (equivalent to DART with λ=0), DART with a same-rank random subspace (Random DART), inference-time steering; post-hoc calibration comparisons are temperature scaling, Platt scaling, beta calibration, isotonic regression. Benchmarks are MMLU, TruthfulQA, AmbigQA.

Overall approach: use difference-in-means to extract an ambiguity direction from ambiguity/disambiguation minimal pairs, verify its selective effect on the model's ambiguity judgments, then use DART fine-tuning to consolidate the calibration improvement into the parameters.

Data and direction extraction

• Data comes from AMBIENT (Liu et al., 2023); each ambiguous sentence is paired with one or more minimally rewritten disambiguated sentences to form contrastive pairs; ambiguity types are annotated according to the taxonomy of Li et al. (2024).
• Only the four categories with sufficient validation samples are kept: pragmatic, lexical, scopal, syntactic; each category uses 80% for training and 20% for validation, and pairs derived from the same ambiguous sentence are placed in the same split. The scale is 907 training pairs and 226 validation pairs (Table 2).
• For each category c and layer ℓ, the candidate direction is the difference of mean paired activations: v_c,ℓ = 1/N_c∑_i (h_ℓ(x^amb_i) − h_ℓ(x^clear_i)), i.e., the difference between the mean activations of ambiguous and disambiguated sentences.

Intervention and layer selection

• Intervention strength is expressed as a relative quantity: the normalized direction is added to the hidden state at a percentage α of that layer's mean activation norm ρℓ, with α taken from 11 levels ranging from 0.1% to 200%; the authors state this allows comparison across layers and models.
• Selectivity is defined as the mean steering effect on ambiguous sentences minus the mean effect on disambiguated sentences, Sel(ℓ, α) = S_amb − S_clear; the ambiguity score ψ(s) is the difference between the model's log probabilities of Yes and No under the ambiguity judgment prompt.
• Layer selection: first average over α, then weight by the number of validation pairs per category to obtain Selw(ℓ), and take arg max to obtain the shared layer ℓ; the authors report that ℓ for all models falls in the relatively early middle-to-lower layers (Figure 2, Appendix E).

DART fine-tuning

• The selected k directions form a matrix V, with projection P = V(VᵀV)⁻¹Vᵀ; the training loss is L_i = (1−λ_i)L_CE,i + λ_i |(I−P)(h_i − h_0,i)|₂², i.e., penalizing activation changes outside the ambiguity subspace while allowing free updates within the subspace.
• The per-sample weight λi equals the current model's probability of the correct answer: samples answered well are more strongly constrained within the subspace, while poorly answered samples focus more on the task objective.
• Implementation details: applied to attention's q_proj and v_proj (Phi-4-mini uses the merged qkv_proj), trained for 3 epochs; LoRA rank r ∈ {4, 8, 16}, learning rate {1e-4, 1e-5}, 6 configurations per model, with the configuration selected by validation-set AUC-EURO.

Confidence acquisition and metrics

• Two protocols: the logits protocol uses the normalized probability over candidate answer labels {A,B,C,D} as confidence (length-normalized likelihood for open-ended generation); the verbalized protocol has the model directly output a confidence number from 0–100, and generations that fail to parse are treated as "incorrect with maximum confidence".
• Metrics: smECE (a calibration error without hyperparameters), AUC-EURO (accounting for both calibration and decision utility), AUROC, accuracy; AmbigQA additionally reports EM and token-level F1.

Experimental Setup

• Models evaluated: gemma-4-E4B-it, Qwen3.5-4B, Qwen3.5-27B, Llama-3.1-8B-Instruct, Llama-3.3-70B-Instruct, OLMo-3-7B-Instruct, Phi-4-mini-instruct, 7 in total; the main text focuses on Llama-3.3-70B-Instruct, the rest are in Appendices G/H.
• Benchmarks: all MMLU subjects; the 684 questions in TruthfulQA MC1 that retain at least 4 options; AmbigQA open generation (5,036 questions evaluated, 5,000 questions fine-tuned).
• Method comparisons: Base, Steering, LoRA, Random DART (same-rank random subspace), DART; post-hoc calibration comparisons are temperature scaling, Platt scaling, beta calibration, isotonic regression.
• Metrics and judgment: smECE, AUC-EURO, AUROC, accuracy; AmbigQA uses EM to judge correctness and reports token-level F1. No LLM judging.
• Statistical tests: paired permutation test (5,000 permutations for MMLU/AmbigQA, 1,000 for TruthfulQA), Holm–Bonferroni correction α=0.05; 95% bootstrap intervals (5,000 resamples).
• Training/selection: LoRA and DART are trained on 5,000 entries of MMLU auxiliary_train; TruthfulQA uses 10-fold cross-fitting repeated 20 times, directly using the adapter trained on MMLU.

Main Results

Llama-3.3-70B-Instruct, logits protocol (per Figure 5 and Table 1):

Swipe horizontally to read the table

Method	MMLU AUC-EURO	MMLU AUROC	TruthfulQA AUC-EURO	TruthfulQA AUROC	AmbigQA smECE↓	AmbigQA AUC-EURO
Base	.753	.617	.710	.631	0.543	0.540
Steering	.770	.766	.745	.726	0.458	0.599
LoRA	.832	.838	.821	.880	0.127	0.777
Random DART	.834	.838	.820	.868	0.136	0.774
DART	.863	.840	.860	.878	0.090	0.778

• The authors state that the main failure mode of the base model is overconfidence, steering brings only limited improvement, and DART aligns confidence with accuracy over a larger confidence range, with little change in task performance.
• The authors state that these trends hold across model families and sizes, and that DART's calibration is consistently better than standard LoRA on multiple-choice QA (Appendix G); in Figure 14, under the logits protocol, many models show statistically significant markers for DART relative to Base.
• On AmbigQA, DART's EM is 0.485 and F1 is 0.596, lower than LoRA's 0.519/0.628 (Table 1), i.e., on open generation the calibration improvement is accompanied by some drop in task metrics.

Random Subspace and Concept Comparisons

• Random-subspace DART: after replacing the ambiguity subspace with a same-rank random subspace, calibration is still better than base, but the authors report that DART with the ambiguity direction has lower calibration error and higher AUC-EURO (Figure 5, Figure 21); the authors therefore argue that the ambiguity subspace itself contributes.
• Sentiment-direction comparison: steering with a sentiment direction extracted from CAD data has little effect on calibration (Figure 20).
• Random-direction steering: compared with 20 random directions, ambiguity steering outperforms most random directions in some settings (e.g., on MMLU logits protocol 1−smECE it outperforms 12/20), but the authors state that this advantage is not stable across datasets and protocols.

Post-hoc Calibration and Verbalized Protocol

• Post-hoc calibration: the four methods such as temperature scaling greatly reduce calibration error and can be combined with steering/LoRA/DART; the authors state that combining often yields further improvement, but the gains are not stably additive (Figure 6, Appendix H).
• Verbalized protocol: confidence calibration is overall far worse than the logits protocol, and improvements from each method are also smaller; Figure 16 shows steering is close to base, and DART improves AUC-EURO only on MMLU. Llama-3.1-8B-Instruct on AmbigQA after LoRA and DART has a prominent parsing-failure problem (per the paper's rule, treated as wrong answer with highest confidence).

Other Ablations and Analyses

• Layer selection position: the highest-selectivity layer for all models is in the earlier middle-lower layers (Figure 11, e.g., ℓ*=30/80 layer for Llama-3.3-70B).
• Token position: the position of peak selectivity varies by model and token; the authors fix offset 1 because it falls within the middle-layer high-selectivity band for all models (Figure 12, 13).
• Qualitative examples: after adding the direction to an MMLU example, the model output mentions potential ambiguity, and after subtracting the direction the output states there is no ambiguity (Figure 3, Table 5); the authors note they do not judge the quality of these outputs, nor do they claim the resulting direction is the true representation of ambiguity.
• Hyperparameter selection: Llama-3.3-70B under the MMLU logits protocol selects the scopal direction, strength 200%, and LoRA/DART rank both 8 (Table 3).

Limitations and Future Directions Noted by the Authors

• The operationalization of ambiguity covers only the component of "an input having multiple interpretations" and does not consider ambiguity in the sense of open-domain question vagueness (Limitations section).
• DART's positive results do not necessarily mean what is recovered is the representation of ambiguity itself; the authors state that interpretability conclusions about whether LMs cleanly represent ambiguity can only be regarded as preliminary (Limitations section).
• No explicit comparison of the three representations—ambiguity, uncertainty, and difficulty; the authors believe the three may mediate confidence in different ways and that combining them may yield greater improvement, but do not prove that the ambiguity representation is necessarily better than other concept representations (Limitations section).
• The authors explain why they did not study uncertainty and difficulty representations: isolating these representations is more difficult and may depend more on the specific model being evaluated (Limitations section).
• The authors plan to release data and code after deanonymization (footnote 2).

Scope of Experimental Coverage

• The models evaluated are 7 instruction-tuned models: gemma-4-E4B-it, Qwen3.5-4B/27B, Llama-3.1-8B-Instruct, Llama-3.3-70B-Instruct, OLMo-3-7B-Instruct, Phi-4-mini-instruct (§4.1).
• The benchmarks are MMLU, TruthfulQA (684 questions), AmbigQA (5,036 questions evaluated); the ambiguity direction data are 907 training pairs and 226 validation pairs converted by AMBIENT, only four ambiguity categories (Table 2).
• The main text results focus on Llama-3.3-70B-Instruct, with results for the other 6 models in Appendices G/H; the authors state that trends are broadly consistent across models.
• The confidence protocols are logits and verbalized; the main evaluation focuses on the logits protocol, with verbalized results in Appendix G.
• The paper does not report complete numerical tables for each method on all models (appendix results are given as figures with error bars, with most values unlabeled).

An interpretability-driven calibration paper: it finds a linear direction of input ambiguity from LM activations, then uses it to constrain fine-tuning to improve confidence estimation.

• Problem: LM confidence cannot accurately predict its correctness; the authors hypothesize that part of the reason is that the model does not exploit its internal encoding of input ambiguity (aleatoric uncertainty).
• Method: on AMBIENT's ambiguity/disambiguation minimal pairs, extract directions via difference-in-means across the four categories pragmatic, lexical, scopal, syntactic, and select layers by selectivity ("effect on ambiguous sentences greater than on disambiguated sentences"); propose DART—during LoRA fine-tuning, penalize activation changes outside the ambiguity subspace, with the regularization weight adjusted per sample according to the model's confidence in the correct answer.
• Direction validation: the highest-selectivity layer for all tested models is in the relatively early middle-to-lower layers; in qualitative examples, adding or subtracting the direction respectively makes the model output mention or deny ambiguity (Figure 3).
• Main results (Llama-3.3-70B-Instruct, logits protocol): on MMLU, AUC-EURO rises from .753 for Base to .863 for DART, and on TruthfulQA from .710 to .860 (Figure 5); on AmbigQA, smECE drops from 0.543 to 0.090 (Table 1). The authors state that accuracy is essentially unchanged on multiple-choice tasks, and the trend is broadly consistent across 7 models.
• Controls: DART with a same-rank random subspace also improves calibration but less than the ambiguity direction; sentiment-direction steering barely affects calibration; under the verbalized protocol, improvements are smaller for all methods.
• Authors' conclusion: DART encodes the calibration improvement into the parameters and removes the need for inference-time intervention, but cannot replace manual risk assessment; the authors emphasize that their results do not mean what is recovered is the true representation of ambiguity, and they did not compare against uncertainty or difficulty representations.