In machine learning, model calibration and predictive inference are essential for producing reliable predictions and quantifying uncertainty to support decision-making.

Optimizing proper loss functions is popularly believed to yield predictors with good calibration properties; the intuition being that for such losses, the global optimum is to predict the ground-truth probabilities, which is indeed calibrated.

Section A.2 shows results using a chat-style verbalized numeric Section A.3 shows results on four extra benchmark tasks made available with Finally, Section A.5 presents and discusses results on feature In this section, we evaluate risk score calibration on the income prediction task across different subpopulations, such as typically done as part of a fairness audit. Figures A1-A2 show group-conditional calibration curves for all models on the ACSIncome task, evaluated on three subgroups specified by the race attribute in the ACS data. We show the three race categories with largest representation. The'Mixtral 8x22B' and'Yi 34B' models shown are the worst offenders, where samples belonging to the'Black' population see consistently lower scores for the same positive label probability when compared to the'Asian' or'White' populations. On average, the'Mixtral 8x22B (it)' model classifies a Black individual with a In fact, this score bias can be reversed for some base models, overestimating scores from Black individuals compared with other subgroups.

A key challenge in probabilistic regression is ensuring that predictive distributions accurately reflect true empirical uncertainty. Minimizing overall prediction error often encourages models to prioritize informativeness over calibration, producing narrow but overconfident predictions. However, in safety-critical settings, trustworthy uncertainty estimates are often more valuable than narrow intervals. Realizing the problem, several recent works have focused on post-hoc corrections; however, existing methods either rely on weak notions of calibration (such as PIT uniformity) or impose restrictive parametric assumptions on the nature of the error. To address these limitations, we propose a novel nonparametric re-calibration algorithm based on conditional kernel mean embeddings, capable of correcting calibration error without restrictive modeling assumptions. For efficient inference with real-valued targets, we introduce a novel characteristic kernel over distributions that can be evaluated in $\mathcal{O}(n \log n)$ time for empirical distributions of size $n$. We demonstrate that our method consistently outperforms prior re-calibration approaches across a diverse set of regression benchmarks and model classes.

In this paper, we introduce a generalization of the standard Stackelberg Games (SGs) framework: Calibrated Stackelberg Games (CSGs) .

In this paper, we introduce a generalization of the standard Stackelberg Games (SGs) framework: Calibrated Stackelberg Games (CSGs) .