Scoring rules elicit probabilistic predictions from a strategic agent by scoring the prediction against a ground truth state. A scoring rule is proper if, from the agent's perspective, reporting the true belief maximizes the expected score. With the development of language models, Wu and Hartline (2024) proposes a reduction from textual information elicitation to the numerical (i.e. probabilistic) information elicitation problem, which achieves provable properness for textual elicitation. However, not all proper scoring rules are well aligned with human preference over text. Our paper designs the Aligned Scoring rule (ASR) for text by optimizing and minimizing the mean squared error between a proper scoring rule and a reference score (e.g. human score). Our experiments show that our ASR outperforms previous methods in aligning with human preference while maintaining properness.
Deep neural networks have demonstrated remarkable performance across numerous learning tasks but often suffer from miscalibration, resulting in unreliable probability outputs. This has inspired many recent works on mitigating miscalibration, particularly through post-hoc recalibration methods that aim to obtain calibrated probabilities without sacrificing the classification performance of pre-trained models. In this study, we summarize and categorize previous works into three general strategies: intuitively designed methods, binning-based methods, and methods based on formulations of ideal calibration. Through theoretical and practical analysis, we highlight ten common limitations in previous approaches. To address these limitations, we propose a probabilistic learning framework for calibration called h-calibration, which theoretically constructs an equivalent learning formulation for canonical calibration with boundedness. On this basis, we design a simple yet effective post-hoc calibration algorithm. Our method not only overcomes the ten identified limitations but also achieves markedly better performance than traditional methods, as validated by extensive experiments. We further analyze, both theoretically and experimentally, the relationship and advantages of our learning objective compared to traditional proper scoring rule. In summary, our probabilistic framework derives an approximately equivalent differentiable objective for learning error-bounded calibrated probabilities, elucidating the correspondence and convergence properties of computational statistics with respect to theoretical bounds in canonical calibration. The theoretical effectiveness is verified on standard post-hoc calibration benchmarks by achieving state-of-the-art performance. This research offers valuable reference for learning reliable likelihood in related fields.
Probabilistic forecasts are typically obtained using state-of-the-art statistical and machine learning models, with model parameters estimated by optimizing a proper scoring rule over a set of training data. If the model class is not correctly specified, then the learned model will not necessarily issue forecasts that are calibrated. Calibrated forecasts allow users to appropriately balance risks in decision making, and it is particularly important that forecast models issue calibrated predictions for extreme events, since such outcomes often generate large socio-economic impacts. In this work, we study how the loss function used to train probabilistic forecast models can be adapted to improve the reliability of forecasts made for extreme events. We investigate loss functions based on weighted scoring rules, and additionally propose regularizing loss functions using a measure of tail miscalibration. We apply these approaches to a hierarchy of increasingly flexible forecast models for UK wind speeds, including simple parametric models, distributional regression networks, and conditional generative models. We demonstrate that state-of-the-art models do not issue calibrated forecasts for extreme wind speeds, and that the calibration of forecasts for extreme events can be improved by suitable adaptations to the loss function during model training. This, however, introduces a trade-off between calibrated forecasts for extreme events and calibrated forecasts for more common outcomes.
We present a general purpose probabilistic forecasting framework, ProbHardE2E, to learn systems that can incorporate operational/physical constraints as hard requirements. ProbHardE2E enforces hard constraints by exploiting variance information in a novel way; and thus it is also capable of performing uncertainty quantification (UQ) on the model. Our methodology uses a novel differentiable probabilistic projection layer (DPPL) that can be combined with a wide range of neural network architectures. This DPPL allows the model to learn the system in an end-to-end manner, compared to other approaches where the constraints are satisfied either through a post-processing step or at inference. In addition, ProbHardE2E can optimize a strictly proper scoring rule, without making any distributional assumptions on the target, which enables it to obtain robust distributional estimates (in contrast to existing approaches that generally optimize likelihood-based objectives, which are heavily biased by their distributional assumptions and model choices); and it can incorporate a range of non-linear constraints (increasing the power of modeling and flexibility). We apply ProbHardE2E to problems in learning partial differential equations with uncertainty estimates and to probabilistic time-series forecasting, showcasing it as a broadly applicable general setup that connects these seemingly disparate domains.
We study how partial information about scoring rules affects fairness in strategic learning settings. In strategic learning, a learner deploys a scoring rule, and agents respond strategically by modifying their features -- at some cost -- to improve their outcomes. However, in our work, agents do not observe the scoring rule directly; instead, they receive a noisy signal of said rule. We consider two different agent models: (i) naive agents, who take the noisy signal at face value, and (ii) Bayesian agents, who update a prior belief based on the signal. Our goal is to understand how disparities in outcomes arise between groups that differ in their costs of feature modification, and how these disparities vary with the level of transparency of the learner's rule. For naive agents, we show that utility disparities can grow unboundedly with noise, and that the group with lower costs can, perhaps counter-intuitively, be disproportionately harmed under limited transparency. In contrast, for Bayesian agents, disparities remain bounded. We provide a full characterization of disparities across groups as a function of the level of transparency and show that they can vary non-monotonically with noise; in particular, disparities are often minimized at intermediate levels of transparency. Finally, we extend our analysis to settings where groups differ not only in cost, but also in prior beliefs, and study how this asymmetry influences fairness.
Variational Autoencoders (VAEs) typically rely on a probabilistic decoder with a predefined likelihood, most commonly an isotropic Gaussian, to model the data conditional on latent variables. While convenient for optimization, this choice often leads to likelihood misspecification, resulting in blurry reconstructions and poor data fidelity, especially for high-dimensional data such as images. In this work, we propose EnVAE, a novel likelihood-free generative framework that has a deterministic decoder and employs the energy score--a proper scoring rule--to build the reconstruction loss. This enables likelihood-free inference without requiring explicit parametric density functions. To address the computational inefficiency of the energy score, we introduce a fast variant, FEnVAE, based on the local smoothness of the decoder and the sharpness of the posterior distribution of latent variables. This yields an efficient single-sample training objective that integrates seamlessly into existing VAE pipelines with minimal overhead. Empirical results on standard benchmarks demonstrate that EnVAE achieves superior reconstruction and generation quality compared to likelihood-based baselines. Our framework offers a general, scalable, and statistically principled alternative for flexible and nonparametric distribution learning in generative modeling.
In this study, we discuss the relationship between two families of density-power-based divergences with functional degrees of freedom -- the H\"{o}lder divergence and the functional density power divergence (FDPD) -- based on their intersection and generalization. These divergence families include the density power divergence and the $\gamma$-divergence as special cases. First, we prove that the intersection of the H\"{o}lder divergence and the FDPD is limited to a general divergence family introduced by Jones et al. (Biometrika, 2001). Subsequently, motivated by the fact that H\"{o}lder's inequality is used in the proofs of nonnegativity for both the H\"{o}lder divergence and the FDPD, we define a generalized divergence family, referred to as the $\xi$-H\"{o}lder divergence. The nonnegativity of the $\xi$-H\"{o}lder divergence is established through a combination of the inequalities used to prove the nonnegativity of the H\"{o}lder divergence and the FDPD. Furthermore, we derive an inequality between the composite scoring rules corresponding to different FDPDs based on the $\xi$-H\"{o}lder divergence. Finally, we prove that imposing the mathematical structure of the H\"{o}lder score on a composite scoring rule results in the $\xi$-H\"{o}lder divergence.
We examine a controlled school choice model where students are categorized into different types, and the distribution of these types within a school influences its priority structure. This study provides a general framework that integrates existing controlled school choice models, including those utilizing reserve rules, quota rules, and bonus-point rules. Specifically, we introduce an adjusted scoring rule that unifies these rules. By achieving a matching that satisfies the stability defined in this framework, matching authorities can effectively manage the trade-offs inherent in controlled school choice markets. Moreover, the priority order for a school is represented as a weak order with each given assignment, meaning that ties are allowed. Our mechanism ensures a stable matching and satisfies strategy-proofness. In particular, when priority orders are restricted to linear orders with each given assignment, our mechanism guarantees student-optimal stability.
Diffusion models generate high-quality synthetic data. They operate by defining a continuous-time forward process which gradually adds Gaussian noise to data until fully corrupted. The corresponding reverse process progressively "denoises" a Gaussian sample into a sample from the data distribution. However, generating high-quality outputs requires many discretization steps to obtain a faithful approximation of the reverse process. This is expensive and has motivated the development of many acceleration methods. We propose to accomplish sample generation by learning the posterior {\em distribution} of clean data samples given their noisy versions, instead of only the mean of this distribution. This allows us to sample from the probability transitions of the reverse process on a coarse time scale, significantly accelerating inference with minimal degradation of the quality of the output. This is accomplished by replacing the standard regression loss used to estimate conditional means with a scoring rule. We validate our method on image and robot trajectory generation, where we consistently outperform standard diffusion models at few discretization steps.
Forecasting is a task that is difficult to evaluate: the ground truth can only be known in the future. Recent work showing LLM forecasters rapidly approaching human-level performance begs the question: how can we benchmark and evaluate these forecasters instantaneously? Following the consistency check framework, we measure the performance of forecasters in terms of the consistency of their predictions on different logically-related questions. We propose a new, general consistency metric based on arbitrage: for example, if a forecasting AI illogically predicts that both the Democratic and Republican parties have 60% probability of winning the 2024 US presidential election, an arbitrageur can trade against the forecaster's predictions and make a profit. We build an automated evaluation system that generates a set of base questions, instantiates consistency checks from these questions, elicits the predictions of the forecaster, and measures the consistency of the predictions. We then build a standard, proper-scoring-rule forecasting benchmark, and show that our (instantaneous) consistency metrics correlate with LLM forecasters' ground truth Brier scores (which are only known in the future). We also release a consistency benchmark that resolves in 2028, providing a long-term evaluation tool for forecasting.
Bayesian reinforcement learning (BRL) is a method that merges principles from Bayesian statistics and reinforcement learning to make optimal decisions in uncertain environments. As a model-based RL method, it has two key components: (1) inferring the posterior distribution of the model for the data-generating process (DGP) and (2) policy learning using the learned posterior. We propose to model the dynamics of the unknown environment through deep generative models, assuming Markov dependence. In the absence of likelihood functions for these models, we train them by learning a generalized predictive-sequential (or prequential) scoring rule (SR) posterior. We used sequential Monte Carlo (SMC) samplers to draw samples from this generalized Bayesian posterior distribution. In conjunction, to achieve scalability in the high-dimensional parameter space of the neural networks, we use the gradient-based Markov kernels within SMC. To justify the use of the prequential scoring rule posterior, we prove a Bernstein-von Mises-type theorem. For policy learning, we propose expected Thompson sampling (ETS) to learn the optimal policy by maximising the expected value function with respect to the posterior distribution. This improves upon traditional Thompson sampling (TS) and its extensions, which utilize only one sample drawn from the posterior distribution. This improvement is studied both theoretically and using simulation studies, assuming a discrete action space. Finally, we successfully extended our setup for a challenging problem with a continuous action space without theoretical guarantees.
In many large-scale classification problems, classes are organized in a known hierarchy, typically represented as a tree expressing the inclusion of classes in superclasses. We introduce a loss for this type of supervised hierarchical classification. It utilizes the knowledge of the hierarchy to assign each example not only to a class but also to all encompassing superclasses. Applicable to any feedforward architecture with a softmax output layer, this loss is a proper scoring rule, in that its expectation is minimized by the true posterior class probabilities. This property allows us to simultaneously pursue consistent classification objectives between superclasses and fine-grained classes, and eliminates the need for a performance trade-off between different granularities. We conduct an experimental study on three reference benchmarks, in which we vary the size of the training sets to cover a diverse set of learning scenarios. Our approach does not entail any significant additional computational cost compared with the loss of cross-entropy. It improves accuracy and reduces the number of coarse errors, with predicted labels that are distant from ground-truth labels in the tree.
Probabilistic predictions are probability distributions over the set of possible outcomes. Such predictions quantify the uncertainty in the outcome, making them essential for effective decision making. By combining multiple predictions, the information sources used to generate the predictions are pooled, often resulting in a more informative forecast. Probabilistic predictions are typically combined by linearly pooling the individual predictive distributions; this encompasses several ensemble learning techniques, for example. The weights assigned to each prediction can be estimated based on their past performance, allowing more accurate predictions to receive a higher weight. This can be achieved by finding the weights that optimise a proper scoring rule over some training data. By embedding predictions into a Reproducing Kernel Hilbert Space (RKHS), we illustrate that estimating the linear pool weights that optimise kernel-based scoring rules is a convex quadratic optimisation problem. This permits an efficient implementation of the linear pool when optimally combining predictions on arbitrary outcome domains. This result also holds for other combination strategies, and we additionally study a flexible generalisation of the linear pool that overcomes some of its theoretical limitations, whilst allowing an efficient implementation within the RKHS framework. These approaches are compared in an application to operational wind speed forecasts, where this generalisation is found to offer substantial improvements upon the traditional linear pool.
When dealing with right-censored data, where some outcomes are missing due to a limited observation period, survival analysis -- known as time-to-event analysis -- focuses on predicting the time until an event of interest occurs. Multiple classes of outcomes lead to a classification variant: predicting the most likely event, a less explored area known as competing risks. Classic competing risks models couple architecture and loss, limiting scalability.To address these issues, we design a strictly proper censoring-adjusted separable scoring rule, allowing optimization on a subset of the data as each observation is evaluated independently. The loss estimates outcome probabilities and enables stochastic optimization for competing risks, which we use for efficient gradient boosting trees. SurvivalBoost not only outperforms 12 state-of-the-art models across several metrics on 4 real-life datasets, both in competing risks and survival settings, but also provides great calibration, the ability to predict across any time horizon, and computation times faster than existing methods.
An automated market maker where the price can cross the zero bound into the negative price domain with applications in electricity, energy, and derivatives markets is presented. A unique feature involves the ability to swap both negatively and positively priced assets between one another, which unlike traditional markets requires a numeraire in the form of a currency. Model extensions to skew and concentrate liquidity are shown. The liquidity fingerprint, payoff, and invariant are compared to the Black-Scholes covered call and the Logarithmic Market Scoring Rule invariants.
Multivariate Gaussian (MVG) distributions are central to modeling correlated continuous variables in probabilistic forecasting. Neural forecasting models typically parameterize the mean vector and covariance matrix of the distribution using neural networks, optimizing with the log-score (negative log-likelihood) as the loss function. However, the sensitivity of the log-score to outliers can lead to significant errors in the presence of anomalies. Drawing on the continuous ranked probability score (CRPS) for univariate distributions, we propose MVG-CRPS, a strictly proper scoring rule for MVG distributions. MVG-CRPS admits a closed-form expression in terms of neural network outputs, thereby integrating seamlessly into deep learning frameworks. Experiments on real-world datasets across multivariate autoregressive and univariate sequence-to-sequence (Seq2Seq) forecasting tasks show that MVG-CRPS improves robustness, accuracy, and uncertainty quantification in probabilistic forecasting.
Neural Stochastic Differential Equations (Neural SDEs) have emerged as powerful mesh-free generative models for continuous stochastic processes, with critical applications in fields such as finance, physics, and biology. Previous state-of-the-art methods have relied on adversarial training, such as GANs, or on minimizing distance measures between processes using signature kernels. However, GANs suffer from issues like instability, mode collapse, and the need for specialized training techniques, while signature kernel-based methods require solving linear PDEs and backpropagating gradients through the solver, whose computational complexity scales quadratically with the discretization steps. In this paper, we identify a novel class of strictly proper scoring rules for comparing continuous Markov processes. This theoretical finding naturally leads to a novel approach called Finite Dimensional Matching (FDM) for training Neural SDEs. Our method leverages the Markov property of SDEs to provide a computationally efficient training objective. This scoring rule allows us to bypass the computational overhead associated with signature kernels and reduces the training complexity from $O(D^2)$ to $O(D)$ per epoch, where $D$ represents the number of discretization steps of the process. We demonstrate that FDM achieves superior performance, consistently outperforming existing methods in terms of both computational efficiency and generative quality.
Probabilistic regression models trained with maximum likelihood estimation (MLE), can sometimes overestimate variance to an unacceptable degree. This is mostly problematic in the multivariate domain. While univariate models often optimize the popular Continuous Ranked Probability Score (CRPS), in the multivariate domain, no such alternative to MLE has yet been widely accepted. The Energy Score - the most investigated alternative - notoriously lacks closed-form expressions and sensitivity to the correlation between target variables. In this paper, we propose Conditional CRPS: a multivariate strictly proper scoring rule that extends CRPS. We show that closed-form expressions exist for popular distributions and illustrate their sensitivity to correlation. We then show in a variety of experiments on both synthetic and real data, that Conditional CRPS often outperforms MLE, and produces results comparable to state-of-the-art non-parametric models, such as Distributional Random Forest (DRF).
This study introduces novel superior scoring rules called Penalized Brier Score (PBS) and Penalized Logarithmic Loss (PLL) to improve model evaluation for probabilistic classification. Traditional scoring rules like Brier Score and Logarithmic Loss sometimes assign better scores to misclassifications in comparison with correct classifications. This discrepancy from the actual preference for rewarding correct classifications can lead to suboptimal model selection. By integrating penalties for misclassifications, PBS and PLL modify traditional proper scoring rules to consistently assign better scores to correct predictions. Formal proofs demonstrate that PBS and PLL satisfy strictly proper scoring rule properties while also preferentially rewarding accurate classifications. Experiments showcase the benefits of using PBS and PLL for model selection, model checkpointing, and early stopping. PBS exhibits a higher negative correlation with the F1 score compared to the Brier Score during training. Thus, PBS more effectively identifies optimal checkpoints and early stopping points, leading to improved F1 scores. Comparative analysis verifies models selected by PBS and PLL achieve superior F1 scores. Therefore, PBS and PLL address the gap between uncertainty quantification and accuracy maximization by encapsulating both proper scoring principles and explicit preference for true classifications. The proposed metrics can enhance model evaluation and selection for reliable probabilistic classification.
Change points in real-world systems mark significant regime shifts in system dynamics, possibly triggered by exogenous or endogenous factors. These points define regimes for the time evolution of the system and are crucial for understanding transitions in financial, economic, social, environmental, and technological contexts. Building upon the Bayesian approach introduced in \cite{c:07}, we devise a new method for online change point detection in the mean of a univariate time series, which is well suited for real-time applications and is able to handle the general temporal patterns displayed by data in many empirical contexts. We first describe time series as an autoregressive process of an arbitrary order. Second, the variance and correlation of the data are allowed to vary within each regime driven by a scoring rule that updates the value of the parameters for a better fit of the observations. Finally, a change point is detected in a probabilistic framework via the posterior distribution of the current regime length. By modeling temporal dependencies and time-varying parameters, the proposed approach enhances both the estimate accuracy and the forecasting power. Empirical validations using various datasets demonstrate the method's effectiveness in capturing memory and dynamic patterns, offering deeper insights into the non-stationary dynamics of real-world systems.