Probabilistic indirect models for undrained shear strength: addressing significant data missing and variability with advanced imputation and machine learning techniques

Accurate prediction of undrained shear strength (su) is crucial for geotechnical design, but is often hampered by substantial uncertainty in traditional empirical methods. This study uses the CLAY/10/7490 global database to develop probabilistic indirect models to predict su based on Atterberg limits and piezocone cone penetration (CPTU) measurements. Firstly, the dataset has a high missing data rate and variability. We test three imputation methods - multivariate normal (MN), multiple imputation by chained equations (MICE), and miss forest (MF) - to fill the missing values. To validate their effectiveness, a Probabilistic Extreme Gradient Boosting (PXGB) model is developed, and the imputation methods are evaluated by comparing the PXGB's performance when trained on the imputed datasets against that on the original incomplete data. Secondly, the indirect model is built by integrating a multi-head attention (MHA) mechanism into an artificial neural network (ANN) to enhance information extraction from limited data, which leads to the MHA-based probabilistic neural networks (MHA-PNN) model. The models' performance, alongside a conventional MN-based prediction model, was evaluated using root mean square error (RMSE), coefficient of determination (R2), mean absolute percentage error (MAPE), conditional interval width (wCI), and coverage rate (CR). Results demonstrate that the proposed MN-enhanced MHA-PNN model substantially outperforms other models in both prediction accuracy and uncertainty quantification. These findings highlight the potential of this integrated strategy for building robust probabilistic indirect models in geotechnical applications, particularly when confronted with sparse and incomplete datasets.

Hybrid Probabilistic Zonotopes for Identifiable and Refinable Predictive Uncertainty

Probabilistic prediction heads in neural networks typically output either a Gaussian mixture or a single conformal region. Neither separates the distinct sources of uncertainty often present in real prediction tasks: a discrete choice among modes, bounded systematic drift within the chosen mode, and irreducible stochastic noise. We introduce the Hybrid Probabilistic Zonotope (HProbZ), an output head that represents these three sources as binary, bounded, and stochastic generators of a zonotope, and admits a closed-form likelihood by convolution. Sharing the bounded generator across prediction steps couples future predictions algebraically, so observing one step refines the predictive distribution at every remaining step in a single forward pass. We establish that the three generators are identifiable from the likelihood up to permutation, and that an HProbZ density is representationally distinct from any finite Gaussian mixture. The same shared structure provides analytic per-mode risk and distribution-free multi-modal conformal sets at inference time. Empirical analysis on representative prediction benchmarks supports the effectiveness of the design relative to same-encoder mixture baselines, while offering structural properties that mixture or convex-conformal predictors do not jointly provide.

Real-time probabilistic tsunami forecasting via generative AI

Explicit onshore tsunami inundation forecasting can improve public risk awareness, but deterministically predicted inundation boundaries under highly uncertain conditions, such as near-field tsunamis generated by megathrust earthquakes, may falsely imply safety outside the boundaries. Consequently, current warnings primarily target coastal tsunami height, not onshore inundation. Although machine learning enables instant inundation predictions, they remain deterministic, lacking uncertainty quantification. Here, we develop a probabilistic ensemble model based on a conditional diffusion model (a type of generative AI) that reconciles accuracy with calibration. Validated with the 2011 Tohoku-oki earthquake data, our model faithfully tracks the postearthquake uncertainty decreasing over time while accurately predicting inundation depth and extent. Our framework shows that generative AI can shift tsunami forecasting from determinism to probabilism, providing a foundation for next-generation early warning.

Conditioning Tree-Based Diffusions and Flows for Probabilistic Tabular Regression

Tree-based diffusion models fit flexible conditional predictive distributions for tabular regression without a neural density estimator, but they inherit their design defaults---noising path, parameterization, training distribution, features, sampler---from the neural setting. We show these defaults are the binding constraint: what a gradient-boosted ensemble actually solves is a supervised regression problem whose conditioning they determine. We present DiffGBM, which makes them explicit along two axes. First, a Gaussian-path flow-matching trainer for $p(y \mid x)$ that learns a velocity field directly and recovers the score algebraically, admitting few-step deterministic ODE sampling. Second, we expose the score-side recipe---residualization, EDM-style preconditioning, log-sigma time sampling, noise-level features, loss weighting, and histogram resolution---as jointly tunable axes over a shared LightGBM surface rather than one frozen bundle. This \emph{score-flex} space represents the published recipe as a special case; across eleven tabular benchmarks under fold-0 tuning, folds-1--5 evaluation, and a matched 40-trial budget and sampler, the selected configurations beat that baseline on \emph{every} dataset (paired Wilcoxon $11/0$, $p<10^{-3}$), with the best aggregate CRPS skill (0.725 vs.\ 0.699) of any row. The two rows are complementary: score-flex buys accuracy with a stochastic sampler and is the slowest row, while flow matching is the cheapest sampler ($5.2\times$ faster than the published baseline) and the best-calibrated DiffGBM row. Tuned non-diffusion baselines still win individual datasets, and stochastic ($\varepsilon>0$) flow samplers do not Pareto-dominate the deterministic corner.

Towards Metric-Agnostic Trajectory Forecasting

Accurate trajectory forecasting of surrounding traffic participants is a core capability for autonomous driving, enabling vehicles to anticipate behavior and plan safe maneuvers. We observe that current state-of-the-art forecasting models on Argoverse 2 and the Waymo Open Motion Dataset tailor their training objectives to the different benchmark metrics. Because these metrics encourage conflicting behavior, we propose a paradigm change for trajectory forecasting: training models with metric-agnostic probabilistic objectives and treating metric optimization as a downstream task applied to the predictive distribution. Concretely, we introduce Trajectory Distribution Evaluation (TraDiE) policies, metric-specific policies that map a predictive distribution to the set of $K$ trajectories and confidences required by trajectory forecasting metrics. We evaluate this framework by introducing DONUT-NLL, which adapts the training objective of the state-of-the-art trajectory forecasting model DONUT to directly optimize the predictive distribution. Using our policies, DONUT-NLL achieves state-of-the-art results on all metrics of the Waymo motion prediction benchmark.

Beyond Point Estimates for Glaucoma Visual Field Forecasting with Diffusion Models

Forecasting visual fields (VFs) is critical for personalized monitoring and treatment planning in glaucoma. This is inherently uncertain due to heterogeneous disease progression and measurement variability, yet most existing methods produce single deterministic predictions that fail to represent this uncertainty. We formulate VF forecasting as a probabilistic prediction problem and the use of conditioned denoising diffusion models to generate distributions of plausible future VFs from longitudinal observations with irregular follow-up intervals. Experiments on two independent VF cohorts show that diffusion-based predictions produce well-calibrated distributions for clinically relevant VF measures. When reduced to a standard point-estimate, the proposed approach achieves state-of-the-art accuracy compared to clinical baselines and prior learning-based methods. Our results highlight the advantages of distributional modeling for VF forecasting and support a shift from point-estimate prediction toward uncertainty-aware, clinically interpretable risk assessment in glaucoma.

A Divergence-Based Method for Weighting and Averaging Model Predictions

This paper uses a minimum divergence framework to introduce a new way of calculating model weights that can be used to average probabilistic predictions from statistical and machine learning models. The method is general and can be applied regardless of whether the models under consideration are fit to data using frequentist, Bayesian, or some other fitting method. The proposed method is motivated in two different ways and is shown empirically to perform better than or on a par with standard model averaging methods, including model stacking and model averaging that relies on Akaike-style negative exponentiated model weighting, especially when the sample size is small. Our theoretical analysis explains why the method has a small-sample advantage.