Numerical results demonstrate significant regret reductions by our method, in comparison with several baselines in a range of multi-armed and linear bandit problems.

Finally,tomaketrainingmorestable,we average the loss over the sequence of predictions for each question to weigh the questions evenly. Is = [0.5, 0.55, ..., 0.95] num_intervals = len(Is) def low_containment_mask(lowers, uppers, labels, Is): # lowers, uppers: Predicted lower and upper bounds of intervals # Is: Target confidence levels # Returns: A list of boolean values indicating which confidence level # has containment ratio below the target level within batch contained = (lowers <= labels) * (labels <= uppers) ratio_contained = contained.mean(dim=0) In total, there are nearly 10,000 questions. Gray text indicates the number of questions after augmenting true/false questions with theirnegations,aprocedureweusetobalancethe dataset. Animportanttaskfor numerical forecasting is outputting calibrated uncertainty estimates.

Theorem 2. Suppose that the predictive distribution Q has the sufficient ability to approximate the true unknown distribution P, given data is i.i.d. Lm(P,Q) = 0 if and only if P = Q when F is a unit ball in a universal RKHS [13]. Becausetheconfidencelevelp2 p1 is exactly equal to the proportion of samples {y1,,yn} covered by the two-sided prediction interval. B.1 Baselines MC-Dropout (MCD) [12]: A variant of standard dropout, named as Monte-Carlo Dropout. Heteroscedastic Neural Network (HNN) [17]: In this approach, similar to a heteroscedastic regression, the network has two outputs in the last layer, corresponding to the predicted mean and variance for each input xi.

Rapidly evolving market conditions call for real-time risk monitoring, but its online estimation remains challenging. In this paper, we study the online estimation of one of the most widely used risk measures, Value at Risk (VaR). Its accurate and reliable estimation is essential for timely risk control and informed decision-making. We propose to use the quantile regression forest in the offline-simulation-online-estimation (OSOA) framework. Specifically, the quantile regression forest is trained offline to learn the relationship between the online VaR and risk factors, and real-time VaR estimates are then produced online by incorporating observed risk factors. To further ensure reliability, we develop a conformalized estimator that calibrates the online VaR estimates. To the best of our knowledge, we are the first to leverage conformal calibration to estimate real-time VaR reliably based on the OSOA formulation. Theoretical analysis establishes the consistency and coverage validity of the proposed estimators. Numerical experiments confirm the proposed method and demonstrate its effectiveness in practice.

During model optimization, the expected calibration error tends to overfit earlier than classification accuracy, indicating distinct optimization objectives for classification error and calibration error. To ensure consistent optimization of both model accuracy and model calibration, we propose a novel method incorporating a probability-dependent gradient decay coefficient into loss function. This coefficient exhibits a strong correlation with the overall confidence level.

Real-world tasks and environments exhibit differences from the static datasets that large language models (LLMs) are typically evaluated on. Such tasks can involve sequential interaction, requiring coherent updating of beliefs in light of new evidence, and making appropriate decisions based on those beliefs. Predicting how LLMs will perform in such dynamic environments is important, but can be tricky to determine from measurements in static settings. In this work, we examine two critical components of LLM performance: the ability of LLMs to coherently update their beliefs, and the extent to which the actions they take are consistent with those beliefs. First, we find that LLMs are largely inconsistent in how they update their beliefs; models can exhibit up to a 30% average difference between the directly elicited posterior, and the correct update of their prior. Second, we find that LLMs also often take actions which are inconsistent with the beliefs they hold. On a betting market, for example, LLMs often do not even bet in the same direction as their internally held beliefs over the underlying outcomes. We also find they have moderate self-inconsistency in how they respond to challenges by users to given answers. Finally, we show that the above properties hold even for strong models that obtain high accuracy or that are well-calibrated on the tasks at hand. Our results highlight the difficulties of predicting LLM behavior in complex real-world settings.