Probabilistic Forecasting

Reinforcement learning with verifiable rewards can in principle train calibrated probabilistic forecasters, since a proper scoring rule such as the Brier score is computed from outcomes alone and is minimized in expectation by the true probability. In practice it degrades calibration, and existing remedies address epistemic uncertainty, where a model's confidence accompanies a verifiably correct or incorrect answer. We study aleatoric forecasting, where the forecast itself is the output and the label is one stochastic outcome, taking NFL in-game win probability as a testbed with the betting market as a reference. Rewarding the realized per-play outcome fails, because the single outcome is a noisy target and the policy gradient corrupts the chain of thought. We introduce a verifiable, label-free reward, a state-conditioned empirical win rate estimated from past outcomes, that removes the label noise, and we keep the gradient off the reasoning, by direct prediction or a gradient mask, so it cannot be corrupted. Trained with this reward alone, without human labels or supervised fine-tuning, a 7B model reaches the calibration of the betting market by direct prediction and is better calibrated than a zero-shot frontier model. That frontier model and a tabular estimator reach the same Brier score as this model, identifying the market's small remaining edge as live in-game information beyond their shared inputs. Masking the gradient, rather than dropping the chain of thought, preserves reasoning from which the forecast follows, which ordinary chain-of-thought training corrupts.

Decision-makers routinely rely on expert judgments accompanied by written explanations, yet explanation quality is difficult to measure at scale. Forecasting tournaments offer a natural testing ground: probabilistic judgments are paired with natural-language rationales and scored against realized outcomes. We introduce Explanation Quality Markers (EQMs), a set of sixty theory-guided reasoning patterns scored by large language models (LLMs). In a pre-registered analysis of over 55,000 forecast-rationale pairs from a multiyear forecasting tournament, EQMs predict accuracy at both the forecast and forecaster levels, consistently outperforming pre-LLM text-analysis methods. More than 90% of statistically significant pattern-level EQM-accuracy correlations match our directional hypotheses. The signal is asymmetric: EQMs identify likely underperformers more reliably than they distinguish the very best forecasters. Benchmarked against traditional indicators of forecasting skill, EQMs are the strongest predictor at the forecast level and competitive at the forecaster level, though weaker than prior accuracy. Human ratings of rationale quality are less consistently correlated with accuracy and place disproportionate weight on rationale length. Results transfer to an independent forecasting study. EQMs provide a scalable, interpretable method for extracting judgment-relevant information from written explanations.

Probabilistic downscaling models the conditional distribution of high-resolution fields given coarse inputs and is a core challenge in atmospheric science, climate modeling, and other multiscale physical systems. A widely used paradigm decomposes the problem into a deterministic mean predictor followed by a stochastic residual generator. While effective in idealized settings, this mean-residual approach frequently produces biased and underdispersive ensembles in real-world applications. We identify a fundamental source of this failure: residual target misspecification, where the residual distribution induced during training systematically differs from the correction distribution required at test time, a mismatch amplified by downscaling bias. We theoretically connect this misspecification to underdispersion through the spread-skill ratio and show that conditional residual distribution matching controls both residual-energy and conditional-mean mismatch. Motivated by this analysis, we introduce ReMatch (Residual Distribution Matching), which aligns training residual targets toward a held-out calibration regime via optimal transport. On a controlled synthetic benchmark with varying bias levels and a real-world HRRR-ERA5 wind-field downscaling task, ReMatch substantially reduces underdispersion, improves calibration, and outperforms strong mean-residual and super-resolution baselines. Our code is available at https://github.com/sdean-group/ReMatch.git.

Earth Observation (EO) forecasting aims to predict future Earth surface dynamics from satellite observations under changing meteorological conditions. In this paper, we view this task as a partially observed, weather-driven world modeling problem, in which weather acts as a conditioning signal, while forecasting remains uncertain due to sparse observations and unobserved land-surface states. However, existing methods do not fully capture this setting: deterministic models collapse uncertainty into a single future prediction, while diffusion-based methods typically treat weather variables as undifferentiated conditioning signals, and existing benchmarks focus mainly on reconstruction accuracy rather than whether forecasts respond correctly to changed weather forcing.We introduce EO-WM, a video diffusion transformer for multispectral EO forecasting. EO-WM incorporates a physically informed conditioning framework that represents meteorological forcing through a climatological baseline, weather anomalies, and cumulative physical stress signals. Specifically, it separates baseline and anomaly through distinct conditioning pathways, and accumulates anomalous forcing over time to capture sustained heat and drought stress. To evaluate weather-response behavior beyond standard metrics, we introduce two diagnostic benchmarks: an Extreme Summer Benchmark for severity-aware prediction of vegetation degradation under extreme weather, and a Seasonal Matched-Pair Benchmark for testing response fidelity under changed weather forcing. Experiments show that EO-WM reduces the error in predicted Normalized Difference Vegetation Index (NDVI) decline amplitude by a relative 5.63% and improves directional hit rate by a relative 7.80%, while remaining competitive on standard pixel-level metrics. The benchmarks and model will be made open-source at https://github.com/Luo-Z13/EO-WM.

Trajectory forecasting for autonomous driving has advanced rapidly, yet representative models often produce uninformative posteriors over forecast modes, causing problems for mode pruning. We trace this to a modeling-training mismatch: forecasters are typically modeled as conditional Gaussian mixture models (GMMs) but trained with a winner-take-all (WTA) loss that assigns each sample to its nearest mode. We argue that this K-means-like hard assignment (one-hot), while preventing mode collapse, is the source of uninformative mode probabilities: it over-segments the trajectory space, ignores relatedness among nearby modes, and yields assignment instability under small perturbations. Guided by this lens, we introduce two post-hoc treatments: (1) test-time posterior-weighted merging that aggregates nearby candidate trajectories; and (2) a one-step expectation-maximization (EM) update that replaces hard labels with soft responsibilities, sharing probability mass across neighboring modes. Across several WTA-trained architectures, these lightweight steps produce more informative, faithfully ranked mode posteriors and strengthen final forecasts on popular displacement metrics -- without retraining. Our analysis unifies recent design choices through a GMM-vs-K-means perspective and offers principled, practical corrections that better align training objectives with inference.

State-of-the-art medium-range AI weather models can outperform traditional Numerical Weather Prediction (NWP) but require massive training budgets. This restricts usage for under-resourced groups and severely limits fast model iteration. Here we develop Otter Weather, a highly efficient spatiotemporal forecasting model designed to democratise high-performance weather prediction with AI. Evaluated on ERA5 reanalysis data at 1.5° resolution using standard WeatherBench protocols, the Otter family significantly advances the skill-compute Pareto frontier. The deterministic version outperforms the best NWP baseline by 9.6% at a 24-hour lead time while requiring fewer than 3.5 A100-days for training. It provides a 2x efficiency gain over lightweight AI models and a 100-fold reduction in compute compared to resource-intensive frontier architectures. We extend these efficiency gains into probabilistic forecasting by training via the Continuous Ranked Probability Score (CRPS). Scaling to a larger architecture, Otter-XL achieves a 9.7% CRPS improvement over the IFS ENS baseline. This yields an almost two-fold increase in predictive skill over comparable lightweight models at similar compute budgets. Otter-XL also outperforms frontier architectures like GenCast by over 2%, while using an order of magnitude less compute. Finally, Otter is applied out-of-the-box to a complex acoustic scattering PDE task where it outperforms a state-of-the-art foundation modelling approach, suggesting that the advances made here might apply across a range of scientific domains.

Sea state prediction is essential for operational maritime applications and coupled earth system modeling, yet current spectral wave models remain computationally prohibitive for many use cases, including online coupling to climate simulations and making probabilistic (ensemble-based) predictions. While deep learning has recently demonstrated strong performance in weather forecasting, existing AI-based wave models are predominantly deterministic and largely limited to bulk variables such as significant wave height, leaving probabilistic sea state estimation largely unexplored. In this work, we propose a diffusion-based generative model for global sea state estimation that conditions on a relatively long history (5 days) of global wind forcing. This generative model directly samples the complex conditional distribution of sea state without autoregressive time-stepping. Unlike prior approaches, our framework naturally extends beyond bulk variables to estimate partition-related variables and derived quantities, such as Stokes drift and mean square slope. Trained on a 30-year global WAVEWATCH-III hindcast, the model achieves substantial computational acceleration compared with numerical spectral models while delivering skillful predictions and a calibrated ensemble spread for the bulk variables. Our results suggest that diffusion-based sea state sampling offers a promising path toward probabilistic wave forecasting and efficient coupling of sea state information into broader earth system models.

Mean Field Variational Inference (MFVI) is widely understood to underestimate posterior variance. By analysing conjugate Bayesian Linear Regression (BLR), we show that this characterization is incomplete: while MFVI underestimates the variance in parameter space, it can overestimate the predictive variance compared to the exact posterior. We show that if the MFVI posterior underestimates predictive variances in some directions, it necessarily overestimates them in others. Crucially, this overestimation occurs in directions where the training data concentrates. This leads to the surprising result that, for a test point drawn from the training distribution, MFVI's expected predictive variance exceeds that of the exact posterior. We demonstrate a pathological case of this effect, where the MFVI posterior fails to reduce predictive variance compared to the prior on in distribution data. We connect these results to the Cold Posterior Effect, arguing that varying the temperature can correct this overestimation, yielding predictions closer to those of the exact posterior. We validate our theory on synthetic and real-world regression tasks.

We study the problem of model selection among probabilistic forecasting models evaluated on datasets of multiple time series. The performance of a model on a single time series is quantified by the average value (score) of a proper scoring rule over a test set, but extending model selection to data sets of time series requires aggregating these scores. Common approaches either rely on scaling scores and averaging them (mean scaled score) or avoid scaling by using alternative statistics such as mean ranks or win rates. However, these approaches can yield conflicting conclusions. We show that such discrepancies arise from the skewness of the distribution of the scores, which is particularly pronounced when test sets are short. The skewness can cause non-mean criteria (e.g., mean rank, median, win rate) to select misspecified models. In contrast, the mean score is immune from this problem. We further show that, as the size of the test sets increases, all aggregation criteria converge to the same model selection decision, mitigating these discrepancies. Our experiments on intermittent demand time series, including data from the M5 competition, highlight the importance of sufficiently large test sets; the mean scaled score appears to be the more reliable approach, also because empirically we found its decision to remain consistent when different scaling factors are adopted.

Time series forecasting is a fundamental machine learning task. Recent work has explored Large Language Models (LLMs) for this purpose due to their strong generalization, pattern recognition, and zero-shot or few-shot capabilities. Despite their suitability for long-context learning, LLMs face challenges in multimodal settings: they lack calibrated probabilistic modeling for non-text data and struggle to align heterogeneous representations. To address these issues, we propose a new framework Diffusion-LLM that integrates a conditional diffusion model into an LLM-based forecasting pipeline. This joint design enables learning the conditional distribution of future data while improving semantic alignment in a shared latent space. We evaluate Diffusion-LLM on six long-term forecasting benchmarks, including ETT, Weather, and ECL. Our method consistently outperforms existing LLM-based baseline, achieving notable gains in ultra-long-term and few-shot forecasting and demonstrating the value of distribution-aware regularization for enhancing robustness and generalization in time series LLMs.

Epidemic forecasting models typically rely on surveillance data reported over administrative regions, treating them as atomic units, thereby obscuring sub-regional spatial structure that shapes disease dynamics. We introduce a spatially structured multimodal epidemic forecasting setting that integrates region-level temporal surveillance data with spatially localized auxiliary signals that are misaligned in resolution and structure, reflecting realistic public health reporting constraints. Building on this formulation, we propose M-SPICE (Multimodal SPatIal Context for Epidemic Forecasting), a structure-aware spatiotemporal forecasting framework that performs joint reasoning over temporal disease dynamics and spatial context via attention-based multimodal fusion, allowing spatial signals to selectively condition temporal representations across forecast horizons. We evaluate our approach on real-world COVID-19, influenza, and influenza-like illness (ILI) forecasting tasks under realistic real-time evaluation protocols. Across all forecasting settings, our method consistently outperforms state-of-the-art multivariate time-series, multimodal, and epidemiological forecasting baselines while maintaining strong probabilistic forecasting performance. Finally, interpretability analyses reveal when, where, and how spatial signals are leveraged, highlighting settings in which purely temporal, region-aggregated models are most likely to fail.

Vegetation monitoring under climate stress requires answering not only how it will evolve given the expected weather, but how it would respond to alternative meteorological conditions. Forecasting models return the expected vegetation state for the observed weather and cannot answer these scenario-conditioned questions, because future weather is fixed to the recorded trajectory. We present VegSim, a geospatial world model for scenario-conditioned vegetation simulation. VegSim infers a latent vegetation state from sparse satellite-derived NDVI histories, past meteorological covariates, and static spatial context, propagates it forward under future weather forcing through recurrent latent dynamics, and decodes predictive NDVI quantiles at each lead time. Because future forcing enters as a controllable input, the same trained model supports probabilistic forecasting under observed weather and conditional simulation under user-defined meteorological forcing, without supervision on scenario responses. We evaluate VegSim on GreenEarthNet across in-distribution data and spatial, temporal, and joint spatial-temporal shift, where it achieves strong point and probabilistic accuracy against time series and Earth observation forecasting baselines while using a compact architecture. We then simulate vegetation responses across Europe under four meteorological scenarios, and in a France summer 2022 case study, obtaining spatially coherent patterns consistent with known sensitivity to temperature and precipitation. The code is available at https://github.com/arco-group/vegsim.

Long-horizon spacecraft trajectory forecasting suffers from error accumulation due to the absence of corrective observations in the forecast regime, making reliable uncertainty estimation crucial for safety-critical decision-making such as space domain awareness and conjunction assessment. While high-fidelity physics-based orbit propagators provide accurate deterministic forecasts, they typically lack calibrated uncertainty estimates over long horizons. We introduce a Predictor--Corrector framework in which a physics-based continuous-time $\textit{deterministic}$ forecaster is augmented with a learned continuous-time $\textit{probabilistic}$ Corrector that models forecast errors. The proposed Corrector can be wrapped around an existing deterministic propagator to improve forecast accuracy while producing sharp and calibrated full-covariance uncertainty estimates. The Corrector is based on Latent Neural Controlled Differential Equations (Latent NCDEs) and models the probabilistic temporal evolution of forecast errors in continuous time, naturally supporting irregular sampling and missing features. We further introduce a loss function that promotes calibration and sharpness in long-horizon uncertainty propagation. We evaluate the proposed framework on long-horizon spacecraft trajectory forecasting using real-world data from NASA's Crustal Dynamics Data Information System (CDDIS), wrapping the Corrector around NASA's General Mission Analysis Tool (GMAT). Across forecast horizons of 2--4 days without observations and six rolling test windows, the proposed approach consistently improves accuracy and uncertainty calibration compared to deterministic baselines and Latent ODE-based correctors, demonstrating the effectiveness of the continuous-time probabilistic Corrector for trajectory forecasting.

Online learning in non-stationary streams is often formulated as tracking a point estimate, although many applications require predicting the full data-generating distribution. We study online distributional prediction under drift and bounded causal adversarial corruption. Candidate laws are represented by variable-size latent cluster configurations that induce predictive distributions. A Gibbs quasi-posterior over configurations yields an online predictor by posterior averaging and can be sampled using reversible-jump MCMC. The method uses no stream likelihood but specifies a kernel-mixture predictive family. We evaluate cumulative Wasserstein-1 regret against the time-varying true law and study periodic restart, sliding-window, and exponential-discounting memory against full history. The bounds separate PAC-Bayesian complexity, corruption-sensitive posterior perturbation, and drift through $A_T^{\mathrm{OT}}=\sum_{t=2}^T W_2^2(p_{t-1}^*,p_t^*)$. Under bounded support, latent quadratic growth, predictive-map regularity, and localized-comparator assumptions, restart has sublinear high-probability cumulative regret when $0\le a<1$, $q,ζ,b<1$, $0\leγ<β<1$, and $0<h<(1-a)/2$. Here $a,q,ζ,b$ control the polynomial growth of drift, predictive-family approximation error, off-order posterior mass, and oracle-order switch cost; $λ_s\asymp s^{-β}$ and $H\asymp T^h$. Corruptions satisfy the segment-local condition $Λ_{r,s}=O(s^γ)$, so a sublinear global budget alone is insufficient. The regime requires oracle-informed tuning, $\varepsilon_T=o(1)$, and fixed confidence or $\log(1/δ_0)=T^{o(1)}$. Sliding-window and discounted guarantees additionally require the stated localized-memory condition.

Forecasting benchmarks for general-purpose AI systems usually inherit the constraints of the real world: outcomes resolve slowly, tail events are rare, and counterfactual questions are difficult to score. We introduce ForecastBench-Sim, a simulated-world forecasting benchmark built on game rollouts from Freeciv, a turn-based strategy game modelled on the Civilization series. Forecasters receive a fixed world report (a structured snapshot of the current game state) and answer questions about hidden future states; the benchmark then continues the simulation and scores forecasts. Because the world is simulated, the same setup can generate continuous or binary forecasting questions at arbitrary time horizons, paired intervention worlds for conditional or causal questions, and resolved examples of rare or disruptive outcomes. We describe the benchmark pipeline, question families, scoring protocol, and release artifacts, and report validation slices from model evaluations and an anonymized human pilot. ForecastBench-Sim is intended to complement real-world forecasting benchmarks by providing controlled, immediately resolvable tasks for studying probabilistic reasoning under dynamic world states.

U-calibration studies online forecasting algorithms whose predictions can be consumed by any unknown downstream agent, guaranteeing sublinear regret simultaneously for all proper loss functions. Existing U-calibration algorithms achieve worst-case optimal $O(\sqrt{T})$ regret for every bounded proper loss, but they fail to adapt to easier losses: as we show, even for smooth losses such as squared loss, they incur $Ω(\sqrt{T})$ regret instead of the optimal $O(\log T)$ regret. In this work, we show that this limitation is not inherent. Specifically, we design a single forecast algorithm that simultaneously achieves $\tilde O(\sqrt{T})$ regret for every bounded proper loss and $O(\log T)$ regret for every bounded smooth proper loss. More generally, our algorithm also attains logarithmic regret for losses that are smooth relative to the log-barrier, which include several non-Lipschitz examples. Our approach is based on a novel variant of Follow-the-Perturbed-Leader (FTPL) in which perturbations are applied directly in the prediction space using self-concordant noise. The resulting analysis also departs substantially from prior FTPL analyses due to the complex nature of this noise and may be of independent interest.

Sparse point observations are increasingly available for precipitation nowcasting, but it is unclear how much they improve dense radar-field forecasts. We partially address this question with a multimodal graph neural network nowcasting system over the Nordic radar domain. The model predicts rain rate every five minutes up to two hours ahead and is trained with different combinations of radar history, MEPS numerical weather prediction, Netatmo surface observations, MSG satellite channels, stochastic noise, and CRPS-based ensemble losses. The study is designed as an ablation of operationally relevant information sources and training objectives. We compare radar-only, NWP-informed, station-informed, satellite-informed, noise-augmented, and CRPS-based configurations using complementary diagnostics on the radar grid, at station locations, for rain onset, and through oracle, displacement, and amplitude scores. The results show that each source improves a different part of the forecast problem. MEPS stabilises radar-only extrapolation, Netatmo observations improve local station and onset diagnostics, and satellite predictors reduce some station-level biases but may activate rain too early when used deterministically. CRPS-based configurations provide the most consistent radar-grid gains, while the combined satellite and CRPS setup gives the best overall oracle/DAS score. These results do not support the conclusion that point observations are uninformative for nowcasting, but they show that local observational skill and spatially coherent radar-field skill are distinct targets. The practical implication is that sparse observations can provide useful local constraints, but their benefit for radar-like fields depends on the training loss, uncertainty representation, and how observation support is encoded in the model.

We use Group Relative Policy Optimization (GRPO), a recently devised sample and memory efficient reinforcement learning method, to finetune pretrained LLMs in the range of 1.5B to 14B parameters equipped with the ability to get current information through the use of a Wikipedia revisions tool, or news summaries, to forecast real events beyond the knowledge cutoff of the LLM, as well as problems made to simulate different aspects of the dynamics of that training. We use the results of these experiments to comment on the scaling capability of LLMs for forecasting, as well as classify how judgmental forecasting fits into the verifiable/unverifiable domain taxonomy, considering the impact of the inherent aleatoric uncertainty when forecasting future events (e.g. the roll of a die). As a result of the GRPO training, we manage to bring a 1.5B parameter transformer (Qwen 2.5 1.5B) to forecasting performance superior to Claude Sonnet 3.5 over the same dataset as measured by cross entropy from the market agreed probabilities. We also discuss various dead ends on the path to this result.

Probabilistic forecasting models are increasingly deployed on multivariate systems with distinct channel physics and operational constraints, but existing benchmarks evaluate neither property at scale. Public canonical multivariate benchmarks cap out at 2,000 channels, while power-system benchmarks either lack temporal structure or probabilistic evaluation. We introduce PowerPhase, a probabilistic forecasting benchmark built on six transmission grids ranging from 2,000 to 36,964 jointly forecasted channels, more than an order of magnitude beyond popular canonical multivariate benchmarks. Each target trajectory is the output of an AC power-flow solve, and PowerPhase ships with constraint-aware metrics, including Safety_mBrier, NECV, and CVaR-alpha, that complement CRPS and Distortion. Across eight baselines and three seeds, distributional accuracy and constraint satisfaction rank models differently, a trade-off we term safety-fidelity. We further propose PowerForge, a scenario-based quantile forecaster with type-specific decoding heads and a causal bridge between variable groups, which achieves the best average rank on every grid.

Two dominant approaches have emerged for generating probabilistic forecasts of physical systems: generative models, such as diffusion or flow matching; and ensembles of deterministic models with stochasticity injected, trained using the continuous ranked probability score (CRPS) loss. While both approaches have demonstrated strong predictive accuracy, the reliability of their uncertainties has not been systematically assessed. We address this gap by developing a framework to evaluate both approaches across diverse 2D spatiotemporal physical systems, under matched model size and computational budget. We assess the reliability of probabilistic emulation by inspecting the empirical coverage of predictive intervals, while also considering accuracy and computational efficiency metrics. CRPS-trained ensembles typically achieve more reliable uncertainties on both single-step prediction and autoregressive rollouts, demonstrating better coverage than the standard alternative of training generative models in a latent space. Moreover, the CRPS approach offers significantly faster inference. When generative models are trained in ambient rather than a compressed latent space, which is often infeasible for high-dimensional problems, they exhibit comparable coverage to CRPS-trained ensembles, though with substantially larger inference latency. In contrast, when CRPS-trained ensembles are trained in latent space they do not show a marked degradation in coverage with respect to ambient space. Both generative models and CRPS-trained ensembles demonstrate good predictive accuracy. To facilitate future research and application, we release AutoCast, a modular framework implementing both generative models and CRPS-trained ensembles, alongside AutoSim, a flexible dataset generation package for rapid prototyping.

Forecasting real-world events requires language-model agents to reason under uncertainty from incomplete, time-bounded information. Yet evaluating whether agents genuinely forecast requires more than final-answer accuracy: a model may be correct by recalling memorized training facts, citing fabricated evidence, or producing an unsupported causal story. We present WorldReasoner, an evaluation framework for temporally valid event forecasting. Each task gives an agent a resolved forecasting question, a simulated forecast date, and access only to evidence available before that date; after resolution, the framework scores the submitted probability, cited evidence, and optional causal event graph. WorldReasoner reports three complementary axes: outcome quality against resolved answers, evidence quality over cited sources, and reasoning quality against post-resolution hindsight graphs. The benchmark is built by an agentic construction pipeline that generates forecasting questions, collects time-stamped evidence, and builds hindsight reference graphs at scale, yielding 345 resolved tasks derived from 14,141 articles with graphs covering 8,087 extracted events. Across six controlled agent settings, temporally valid retrieval is the strongest driver of outcome accuracy; causal graph construction improves key-event recovery; and correct graph-enabled forecasts are more strongly grounded in key events and relevant sources, yet agents still struggle to convert grounded evidence into calibrated probabilities.

We introduce First-Order Trajectory Matching (FTM), a surrogate-modeling method that learns the first-order local transport of probability mass from trajectories of stochastic systems. By matching the symmetric first-order motion of trajectories, FTM learns the probability current velocity, whose flow preserves time marginals to match ensemble averages, while also capturing current-like trajectory quantities such as fluxes, circulations, and barrier-crossing currents. FTM learns the current velocity directly from trajectories, avoiding drift, diffusion, and score estimation. Our stability analysis separates discretization error from sampling variance and shows that the one-step simulation-free FTM loss is stable when temporal resolution and sample size are properly balanced. Across stochastic dynamical systems and PDE examples, we empirically demonstrate that FTM provides trajectory-aware ensemble predictions at low, deterministic-rollout cost.

As renewable energy integration increases market volatility, probabilistic electricity price forecasting has become essential for effective risk management. However, current-proper-scoring rules often prioritize forecast sharpness at the expense of calibration, leading to overconfident and statistically unreliable uncertainty estimates. This work highlights the critical gap between theoretical scoring and practical calibration, demonstrating that models can become mere proxies for deterministic forecasts when reliability is neglected. We conclude that future research must shift toward calibration-aware objectives and architectures to ensure the distributional integrity of energy market forecasts.

Statistical post-processing has proven to be an effective tool in improving ensemble forecast of different weather variables. Case studies show that post-processing can remedy the typically underdispersive and potentially biased behaviour of the ensemble while optimizing a proper scoring rule expressing the forecast skill. The price of these positive effects is generally a deterioration in sharpness; the width of the central prediction intervals and the uncertainty of the predictions are increasing, especially for shorter lead times. This work aims to reduce the extent of the latter phenomenon for neural network-based parametric post-processing methods by extending the network's loss function with a penalty term. We demonstrate the effect of the proposed technique for 2m temperature ensemble forecasts of the European Centre for Medium-Range Weather Forecasts downloaded from the EUPPBench benchmark dataset and verified against synoptic observations. Here, the predictive distribution is Gaussian, and we use the continuous ranked probability score (CRPS) as loss function. The case studies confirm a substantial relative decrease ($8.2\%-12.5\%$) in the width of the nominal central prediction interval compared to the width of the predictive distribution computed without the penalty term, while there is no deterioration in the mean CRPS of probabilistic forecasts and in the RMSE of the predictive mean.

Proper scoring rules provide a rigorous theoretical basis for the training and evaluation of probabilistic forecasts. However, in the presence of right censoring, the event time is only partially observed, rendering conventional scoring rules inapplicable in their standard form. We propose a framework for proper scoring of right-censored survival outcomes based on a simple idea: first, map the predictive distribution through the censoring mechanism, then apply the underlying proper score on the induced observed-data law. This yields localized scores for fixed censoring times and marginalized scores when the censoring time is random or only partially observed. The resulting construction recovers familiar right-censored likelihood and IPCW-type criteria within a coherent framework, while also yielding right-censored versions of the CRPS, pinball loss, Brier score, and energy score. We show that the marginalized score is proper under conditional independent censoring and strictly proper on the identifiable region. The same principle also leads to censored engression, a sample-based learning objective for multivariate right-censored survival modeling. In experiments, our scores correctly rank the oracle forecast across several censoring regimes, whereas forecast-dependent plug-in weighted scores can exhibit ranking reversals. Censored engression likewise substantially improves over naive training on censored outcomes.

Vessel trajectory prediction from Automatic Identification System (AIS) data is essential for maritime situational awareness, yet it remains challenging due to irregular sampling, missing reports, and complex dynamics. Beyond accurate point forecasts, maritime applications also demand well-calibrated uncertainty estimates for reliable decision-making. Bayesian Neural Ordinary Differential Equations (ODEs) offer a principled framework for continuous-time trajectory modeling with uncertainty quantification by placing a prior over the neural vector field parameters. However, the commonly used isotropic Gaussian weight prior fails to encode informative structural properties of vessel dynamics, such as smoothness and locality. Existing function-space Bayesian neural network methods address this limitation for static mappings, but do not transfer directly to Neural ODEs, where the primary quantity of interest is the trajectory rather than the vector field itself. In principle, one could place a Gaussian process (GP) prior directly over ODE solutions, but this requires propagating distributions through a nonlinear ODE solver, which is analytically intractable. To address this challenge, we adopt a practical approach that imposes a GP-kernel-based prior directly on the vector field evaluated at a finite set of measurement points. Specifically, we augment the standard weight-space variational objective with a kernel-based regularizer that penalizes deviations of the vector field from the structure implied by a GP prior. To handle long and irregular AIS trajectories, we further combine this function-space regularization with probabilistic multiple shooting, which decouples inference across temporal segments while maintaining global consistency.

Learning Value-at-Risk (VaR) and Expected Shortfall (ES) is important for managing financial risks effectively. Existing approaches with limited parameters are vulnerable to model misspecification in the era of big data. To address this limitation, we propose a large tail risk model, the retrieval-enhanced self-grouping autoencoder (ReSGA), which is designed with millions of parameters to exploit the rich cross-sectional dependence and long-term temporal dynamics of assets using their characteristics. Applied to monthly US equity returns from 1926 to 2023 with 153 firm characteristics, ReSGA outperforms twelve econometric and machine learning competitors in terms of out-of-sample loss and statistical backtesting. In addition, its forecast advantages can translate into significant economic gains from long-short decile portfolios that are constructed by a new size-enhanced left-side momentum strategy. To clarify the role of complexity, we further conduct a systematic scaling analysis and demonstrate that improvements in joint VaR-ES forecasting are primarily driven by data complexity rather than model complexity. Finally, our analyses of group-importance and transfer-learning exhibit the interpretability and cross-market generalizability of ReSGA.

Studying the spatiotemporal evolution of scientific phenomena often relies on costly simulations and experiments. Machine learning-based surrogate models reduce this cost, but most are limited to deterministic forward prediction. Scientific temporal analysis often requires both forward prediction and backward inference, while temporal evolution is not always uniquely determined, especially for the inverse problem. We introduce DiffUNet^2, a bidirectional conditional diffusion model for probabilistic scientific temporal prediction. It supports both forward and backward prediction within a shared model. We evaluate DiffUNet^2 on four scientific temporal datasets spanning fluid dynamics, chemical reaction dynamics, and material deformation, against deterministic and probabilistic baselines. Results show that DiffUNet^2 achieves strong predictive performance in both temporal directions and high probabilistic ensemble quality compared with existing baselines. To support practical scientific exploration beyond prediction, we further extend DiffUNet^2's generation ability with target-guided state editing, allowing states of interest to be specified and explored in both temporal directions.

Recent advances in machine learning have produced probabilistic weather forecasting models comparable to state-of-the-art numerical weather predictors. But no model consistently dominates spatio-temporally, and relative performance is highly context-dependent. This motivates adaptive methods for combining multiple forecasts to obtain improvements and robustness. While combined forecasts have been proposed in the literature, these are achieved either through supervised learning or through prediction with expert advice methods. We introduce AdaWeather, an adaptive framework that combines many probabilistic forecasts using both machine learning as well as mixture of experts to arrive at a unified improved probabilistic forecast. While traditional expert methods develop the regret bounds with respect to the best single expert in hindsight, we extend the algorithm and analysis to show our method has logarithmic regret compared to the best static mixture of experts in hindsight. Empirically, we focus on forecasting temperature, and observe improvements over existing methods.

Reliable quantification of malaria dynamics in sub-Saharan Africa is hindered by short, noisy, and spatially heterogeneous surveillance records. In Ghana, health-facility data from 2014 to 2023 reveal non-linear and age-specific fluctuations in hospital admissions, yet existing approaches struggle to capture stochastic variability or provide credible uncertainty bounds. This study develops a Bayesian nonlinear inference framework that integrates a cubic baseline with a damped oscillatory kernel, estimated via an affine-invariant ensemble Markov Chain Monte Carlo sampler. The framework accommodates limited data, models parameter uncertainty, and generates probabilistic forecasts for children under five years and individuals aged five years or more. Results show strong empirical adequacy ($R^2 = 0.9958$ for $<5$ years; $R^2 = 0.9956$ for $\geq 5$ years) with residual errors below $2\%$ and well-mixed posteriors confirming convergence. District-level analysis reveals pronounced spatial heterogeneity, with coefficients of variation ranging from $<0.07$ in urban centres such as Kumasi to $>3.3$ in peripheral districts such as Mpohor and Bia East. Forecasts for 2024-2026 indicate a gradual resurgence: from 137,000 to 149,000 cases among children under five years and from 348,000 to 375,000 cases among older individuals, with uncertainty widening over time. By producing probabilistic forecasts, this Bayesian framework provides a principled tool for anticipating malaria fluctuations and strengthening data-driven decision-making in Ghana's national malaria control strategy.