VERDI: Single-Call Confidence Estimation for Verification-Based LLM Judges via Decomposed Inference

Verbalized confidence, long dismissed as overconfident, coarse, and prone to round-number clustering, is now the more robust soft-scoring mechanism for LLM-as-a-Judge on top-tier proprietary models. Across SummEval, AggreFact, and HelpSteer2, spanning up to 18 LLMs, we show that the standard advice to prefer log-probabilities no longer holds on post-2025 models, where verbalized confidence is the better signal. We call this a compatibility shift. On top of a standard verbalized-confidence baseline, we introduce two new ingredients: an overconfidence advisory and self-debate. Together they improve calibration, score-distribution spread, and robustness to task subjectivity. We further observe a generation effect: post-2025 models accommodate these two additions with little balanced-accuracy cost, whereas pre-2025 models pay a measurable penalty. Compared with logprob-based G-Eval, verbalized confidence is the more subjectivity-robust soft signal on GPT-family top-tier releases. The shift is invisible under accuracy-only reporting. Rather than defaulting to hard predictions, we recommend broader use of soft scoring in LLM-as-a-Judge. More broadly, verbalized confidence has moved from a weaker substitute for logprobs to a practical soft-scoring mechanism for contemporary LLM judges.

Reliably knowing when a language model is correct is almost as important as being correct. We introduce prover-verifier deliberation (PVD), an inference-time protocol grounded in interactive proof theory, as a mechanism for selective prediction: the protocol produces both an answer and a structured confidence verdict, allowing a system to report high-confidence answers while abstaining on uncertain cases. In each dialogue, a prover defends a candidate answer through checkable sub-claims while a verifier issues targeted challenges and returns \textsc{Accept}, \textsc{Challenge}, or \textsc{Reject}. Because frozen language models are imperfect provers and verifiers operating over a noisy channel, formal soundness and completeness guarantees do not transfer; instead, we characterize the protocol empirically through its coverage-precision behavior. Our main experiment uses Claude Sonnet 4.6 as prover and Claude Haiku 4.5 as verifier on GPQA Diamond. Questions accepted with no answer revision, which we call Accept + No Change (ANC), are reported as the high-confidence subset; we evaluate this subset by its precision and coverage. ANC separates reliable from unreliable answers, yielding a $\sim$30pp HC-Prec gap over the non-ANC complement. Robustness experiments with GPT and Gemini pairings show that high HC-Prec can transfer across model families, while verifier strictness and domain competence largely determine the size of the selection gap. On Humanity's Last Exam, weaker prover-verifier pairings can collapse or invert the ANC signal, illustrating a practical failure mode when the verifier operates outside its effective region. Comparisons with self-consistency, universal self-consistency, multi-agent debate, and Reflexion suggest that prover-verifier deliberation supplies a distinct argument-defensibility signal for selective prediction.

LLM-as-judge frameworks are increasingly used for automatic NLG evaluation, yet their per-instance reliability remains poorly understood. We present a two-pronged diagnostic toolkit applied to SummEval: $\textbf{(1)}$ a transitivity analysis that reveals widespread per-input inconsistency masked by low aggregate violation rates ($\barρ = 0.8$-$4.1\%$), with $33$-$67\%$ of documents exhibiting at least one directed 3-cycle; and $\textbf{(2)}$ split conformal prediction sets over 1-5 Likert scores providing theoretically-guaranteed $\geq(1{-}α)$ coverage, with set width serving as a per-instance reliability indicator ($r_s = {+}0.576$, $N{=}1{,}918$, $p < 10^{-100}$, pooled across all judges). Critically, prediction set width shows consistent cross-judge agreement ($\bar{r} = 0.32$-$0.38$), demonstrating it captures document-level difficulty rather than judge-specific noise. Across four judges and four criteria, both diagnostics converge: criterion matters more than judge, with relevance judged most reliably (avg. set size $\approx 3.0$) and coherence moderately so (avg. set size $\approx 3.9$), while fluency and consistency remain unreliable (avg. set size $\approx 4.9$). We release all code, prompts, and cached results.