Polynomial chaos expansions (PCE) are widely used for uncertainty quantification (UQ) tasks, particularly in the applied mathematics community. However, PCE has received comparatively less attention in the statistics literature, and fully Bayesian formulations remain rare--especially with implementations in R. Motivated by the success of adaptive Bayesian machine learning models such as BART, BASS, and BPPR, we develop a new fully Bayesian adaptive PCE method with an efficient and accessible R implementation: khaos. Our approach includes a novel proposal distribution that enables data-driven interaction selection, and supports a modified g-prior tailored to PCE structure. Through simulation studies and real-world UQ applications, we demonstrate that Bayesian adaptive PCE provides competitive performance for surrogate modeling, global sensitivity analysis, and ordinal regression tasks.

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We study the sensitivity of a MAP configuration of a discrete probabilistic graphical model with respect to perturbations of its parameters. These perturbations are global, in the sense that simultaneous perturbations of all the parameters (or any chosen subset of them) are allowed. Our main contribution is an exact algorithm that can check whether the MAP configuration is robust with respect to given perturbations. Its complexity is essentially the same as that of obtaining the MAP configuration itself, so it can be promptly used with minimal effort. We use our algorithm to identify the largest global perturbation that does not induce a change in the MAP configuration, and we successfully apply this robustness measure in two practical scenarios: the prediction of facial action units with posed images and the classification of multiple real public data sets. A strong correlation between the proposed robustness measure and accuracy is verified in both scenarios.

In scientific applications, predictive modeling is often of limited use without accurate uncertainty quantification (UQ) to indicate when a model may be extrapolating or when more data needs to be collected. Bayesian Neural Networks (BNNs) produce predictive uncertainty by propagating uncertainty in neural network (NN) weights and offer the promise of obtaining not only an accurate predictive model but also accurate UQ. However, in practice, obtaining accurate UQ with BNNs is difficult due in part to the approximations used for practical model training and in part to the need to choose a suitable set of hyperparameters; these hyperparameters outnumber those needed for traditional NNs and often have opaque effects on the results. We aim to shed light on the effects of hyperparameter choices for BNNs by performing a global sensitivity analysis of BNN performance under varying hyperparameter settings. Our results indicate that many of the hyperparameters interact with each other to affect both predictive accuracy and UQ. For improved usage of BNNs in real-world applications, we suggest that global sensitivity analysis, or related methods such as Bayesian optimization, should be used to aid in dimensionality reduction and selection of hyperparameters to ensure accurate UQ in BNNs.

We present SAInT, a Python-based tool for visually exploring and understanding the behavior of Machine Learning (ML) models through integrated local and global sensitivity analysis. Our system supports Human-in-the-Loop (HITL) workflows by enabling users - both AI researchers and domain experts - to configure, train, evaluate, and explain models through an interactive graphical interface without programming. The tool automates model training and selection, provides global feature attribution using variance-based sensitivity analysis, and offers per-instance explanation via LIME and SHAP. We demonstrate the system on a classification task predicting survival on the Titanic dataset and show how sensitivity information can guide feature selection and data refinement.

We present a neural framework for learning conditional optimal transport (OT) maps between probability distributions. Our approach introduces a conditioning mechanism capable of processing both categorical and continuous conditioning variables simultaneously. At the core of our method lies a hypernetwork that generates transport layer parameters based on these inputs, creating adaptive mappings that outperform simpler conditioning methods. Comprehensive ablation studies demonstrate the superior performance of our method over baseline configurations. Furthermore, we showcase an application to global sensitivity analysis, offering high performance in computing OT-based sensitivity indices. This work advances the state-of-the-art in conditional optimal transport, enabling broader application of optimal transport principles to complex, high-dimensional domains such as generative modeling and black-box model explainability.

This study demonstrates the capabilities of several methods for analyzing the sensitivity of neural networks to perturbations of the input data and interpreting their underlying mechanisms. The investigated approaches include the Sobol global sensitivity analysis, the local sensitivity method for input pixel perturbations and the activation maximization technique. As examples, in this study we consider a small feedforward neural network for analyzing an open tabular dataset of clinical diabetes data, as well as two classical convolutional architectures, VGG-16 and ResNet-18, which are widely used in image processing and classification. Utilization of the global sensitivity analysis allows us to identify the leading input parameters of the chosen tiny neural network and reduce their number without significant loss of the accuracy. As far as global sensitivity analysis is not applicable to larger models we try the local sensitivity analysis and activation maximization method in application to the convolutional neural networks. These methods show interesting patterns for the convolutional models solving the image classification problem. All in all, we compare the results of the activation maximization method with popular Grad-CAM technique in the context of ultrasound data analysis.

Scientific Machine Learning (SciML) is a recently emerged research field which combines physics-based and data-driven models for the numerical approximation of differential problems. Physics-based models rely on the physical understanding of the problem at hand, subsequent mathematical formulation, and numerical approximation. Data-driven models instead aim to extract relations between input and output data without arguing any causality principle underlining the available data distribution. In recent years, data-driven models have been rapidly developed and popularized. Such a diffusion has been triggered by a huge availability of data (the so-called big data), an increasingly cheap computing power, and the development of powerful machine learning algorithms. SciML leverages the physical awareness of physics-based models and, at the same time, the efficiency of data-driven algorithms. With SciML, we can inject physics and mathematical knowledge into machine learning algorithms. Yet, we can rely on data-driven algorithms' capability to discover complex and non-linear patterns from data and improve the descriptive capacity of physics-based models. After recalling the mathematical foundations of digital modelling and machine learning algorithms, and presenting the most popular machine learning architectures, we discuss the great potential of a broad variety of SciML strategies in solving complex problems governed by partial differential equations. Finally, we illustrate the successful application of SciML to the simulation of the human cardiac function, a field of significant socio-economic importance that poses numerous challenges on both the mathematical and computational fronts. The corresponding mathematical model is a complex system of non-linear ordinary and partial differential equations describing the electromechanics, valve dynamics, blood circulation, perfusion in the coronary tree, and torso potential. Despite the robustness and accuracy of physics-based models, certain aspects, such as unveiling constitutive laws for cardiac cells and myocardial material properties, as well as devising efficient reduced order models to dominate the extraordinary computational complexity, have been successfully tackled by leveraging data-driven models.

Explainable Artificial Intelligence (XAI) is central to the debate on integrating Artificial Intelligence (AI) and Machine Learning (ML) algorithms into clinical practice. High-performing AI/ML models, such as ensemble learners and deep neural networks, often lack interpretability, hampering clinicians' trust in their predictions. To address this, XAI techniques are being developed to describe AI/ML predictions in human-understandable terms. One promising direction is the adaptation of sensitivity analysis (SA) and global sensitivity analysis (GSA), which inherently rank model inputs by their impact on predictions. Here, we introduce a novel delta-XAI method that provides local explanations of ML model predictions by extending the delta index, a GSA metric. The delta-XAI index assesses the impact of each feature's value on the predicted output for individual instances in both regression and classification problems. We formalize the delta-XAI index and provide code for its implementation. The delta-XAI method was evaluated on simulated scenarios using linear regression models, with Shapley values serving as a benchmark. Results showed that the delta-XAI index is generally consistent with Shapley values, with notable discrepancies in models with highly impactful or extreme feature values. The delta-XAI index demonstrated higher sensitivity in detecting dominant features and handling extreme feature values. Qualitatively, the delta-XAI provides intuitive explanations by leveraging probability density functions, making feature rankings clearer and more explainable for practitioners. Overall, the delta-XAI method appears promising for robustly obtaining local explanations of ML model predictions. Further investigations in real-world clinical settings will be conducted to evaluate its impact on AI-assisted clinical workflows.