Isotonic regression is a canonical tool for estimating monotone functions and calibrating probabilistic predictors. We provide a fully sharp finite-sample characterization of its worst-case degrees of freedom on binary samples. Specifically, we identify the binary sequences that maximize the number of distinct fitted values produced by isotonic regression. We develop a sharp bound on the degrees of freedom with a leading term of $\frac{3}{(4π^2)^{1/3}} n^{2/3}$ using analytic number theory, improving on previous bounds. We then apply this result to calibration. Calibration is a central requirement for probabilistic prediction, and isotonic regression is a widely used post-processing method for improving calibration. Building on deterministic degrees-of-freedom bounds, we derive, to our knowledge, the first nontrivial distribution-free guarantee on the Expected Calibration Error (ECE) of isotonic regression. This ECE bound is fully model-free and distribution-free, only assuming $Y \in \{0,1\}$.
Reasoning language models are increasingly asked not only to answer difficult questions, but also to estimate their likelihood of success. Existing methods typically elicit confidence only once: either before thinking or after answering. We argue that confidence in reasoning models is state-dependent: before thinking, confidence should estimate the chance of the model correctly solving the prompt, while after thinking it should predict whether the realized answer is likely to be correct. This distinction determines the appropriate supervision target: prompt-level success should supervise confidence estimates made after seeing the prompt, while individual answer-level correctness should supervise confidence estimates made after answering. We introduce CALIBER (Calibration Before and After Reasoning), which elicits both estimates and supervises each with the target matched to its information state. Under this unified protocol, CALIBER reduces Expected Calibration Error (ECE) by 52.5% over the strongest single-confidence baseline on BigMathDigits for the 7B model, while achieving the best Brier score and AUROC, and remains within 2.1 points of the best accuracy. Further, on a larger 30B model, CALIBER achieves the best ECE on BigMathDigits while remaining competitive in Brier score and AUROC. Out of distribution, it achieves the best ECE and Brier score on GPQA and TriviaQA, and remains competitive on SimpleQA. Ablations further show that this position-target alignment is most beneficial under distribution shift where it consistently reduces calibration error across all out-of-distribution benchmarks.
Multimodal Large Language Models (MLLMs) show great potential in medical tasks, but their elicited confidence often misaligns with actual accuracy, potentially leading to misdiagnosis or overlooking correct advice. This study presents the first comprehensive analysis of the relationship between accuracy and confidence in medical MLLMs. It proposes a novel method that combines Multi-Strategy Fusion-Based Interrogation (MS-FBI) with auxiliary expert LLM assessment, aiming to improve confidence calibration in Medical Visual Question Answering (VQA). Experiments demonstrate that our method reduces the Expected Calibration Error (ECE) by an average of 40\% across three Medical VQA datasets, significantly enhancing MLLMs' reliability. The findings highlight the importance of domain-specific calibration for MLLMs in healthcare, offering a more trustworthy solution for AI-assisted diagnosis.