Selection bias, arising from the systematic inclusion or exclusion of certain samples, poses a significant challenge to the validity of causal inference. While Bareinboim et al. introduced methods for recovering unbiased observational and interventional distributions from biased data using partial external information, the complexity of the backdoor adjustment and the method's strong reliance on observational data limit its applicability in many practical settings. In this paper, we formally discover the recoverability of $P(Y^*_{x^*})$ under selection bias with experimental data. By explicitly constructing counterfactual worlds via Structural Causal Models (SCMs), we analyze how selection mechanisms in the observational world propagate to the counterfactual domain. We derive a complete set of graphical and theoretical criteria to determine that the experimental distribution remain unaffected by selection bias. Furthermore, we propose principled methods for leveraging partially unbiased observational data to recover $P(Y^*_{x^*})$ from biased experimental datasets. Simulation studies replicating realistic research scenarios demonstrate the practical utility of our approach, offering concrete guidance for mitigating selection bias in applied causal inference.

Missing data appears when parts of the data are not available for a given variable or a given observation. It is an ubiquitous problem in a wide range of scientific disciplines, including sensor networks, geophysical data analysis, radar and image processing, remote sensing, ecological statistics and biomedical studies, just to name a few [1]-[5]. Signal processing is no exception to the rule, where missing data mainly come from sensor malfunction, hidden or impossible measurements, human errors and natural hazards, all of which can hinder a thorough understanding, analysis, and interpretation of the signal. One of the earliest work on missing data was published in 1932 by Wilks, who mentioned the need to extract as much information as possible from fragmentary answers of questionnaires in social sciences and government statistics. Therefore, it is not surprising that the first discipline to witness this issue was mathematical statistics. This led Wilks to derive efficient estimators for the parameters of a normal bivariate distribution when the data contain missing values [6]. This work was extended to the multivariate case by Lord in 1955 [7]. Since the early 1970's, the literature in missing data has flourished with the development of computational capacity, leading to major developments in signal processing and its related fields, such as statistical inference [2], data analysis [8] and machine learning [9]. In particular, the formulation of a missing-data theory framework by Rubin in [10], which describes the relation between missingness and data values in the so-called missing-data mechanisms, has allowed tremendous advancements in statistical analysis. Therefore, a tutorial paper aiming to summarize the existing and novel strategies in the SP & ML literature addressing various problems related to missing data, such as parameter estimation, matrix completion, missing data imputation and learning with missing values, as well as showing their potential applications, is an urgent desideratum. This tutorial aims to provide practitioners with vital tools, in an accessible way, to answer the question: How to deal with missing data? There are many strategies to handle incomplete signals.

Most expressive variational families -- such as normalizing flows -- lack practical convergence guarantees, as their theoretical assurances typically hold only at the intractable global optimum. In this work, we present a general recipe for constructing tuning-free, asymptotically exact variational flows from involutive MCMC kernels. The core methodological component is a novel representation of general involutive MCMC kernels as invertible, measure-preserving iterated random function systems, which act as the flow maps of our variational flows. This leads to three new variational families with provable total variation convergence. Our framework resolves key practical limitations of existing variational families with similar guarantees (e.g., MixFlows), while requiring substantially weaker theoretical assumptions. Finally, we demonstrate the competitive performance of our flows across tasks including posterior approximation, Monte Carlo estimates, and normalization constant estimation, outperforming or matching No-U-Turn sampler (NUTS) and black-box normalizing flows.

We present a generative learning framework for probabilistic sampling based on an extension of the Probabilistic Learning on Manifolds (PLoM) approach, which is designed to generate statistically consistent realizations of a random vector in a finite-dimensional Euclidean space, informed by a limited (yet representative) set of observations. In its original form, PLoM constructs a reduced-order probabilistic model by combining three main components: (a) kernel density estimation to approximate the underlying probability measure, (b) Diffusion Maps to uncover the intrinsic low-dimensional manifold structure, and (c) a reduced-order Ito Stochastic Differential Equation (ISDE) to sample from the learned distribution. A key challenge arises, however, when the number of available data points N is small and the dimensionality of the diffusion-map basis approaches N, resulting in overfitting and loss of generalization. To overcome this limitation, we propose an enabling extension that implements a synthesis of Double Diffusion Maps -- a technique capable of capturing multiscale geometric features of the data -- with Geometric Harmonics (GH), a nonparametric reconstruction method that allows smooth nonlinear interpolation in high-dimensional ambient spaces. This approach enables us to solve a full-order ISDE directly in the latent space, preserving the full dynamical complexity of the system, while leveraging its reduced geometric representation. The effectiveness and robustness of the proposed method are illustrated through two numerical studies: one based on data generated from two-dimensional Hermite polynomial functions and another based on high-fidelity simulations of a detonation wave in a reactive flow.

Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes, specifically for problems where the data obey a conservation constraint from an underlying partial differential equation. Our approach combines discrete exterior calculus and nonlinear optimal recovery to infer relationships between vertex and edge values. This framework yields data-driven predictions with uncertainty quantification across the entire graph, even when observations are limited to a subset of vertices and edges. By optimizing over the reproducing kernel Hilbert space norm while applying a maximum likelihood estimation penalty on kernel complexity, our method ensures that the resulting surrogate strictly enforces conservation laws without overfitting. We demonstrate our method on two representative applications: subsurface fracture networks and arterial blood flow. Our results show that the method maintains high accuracy and well-calibrated uncertainty estimates even under severe data scarcity, highlighting its potential for scientific applications where limited data and reliable uncertainty quantification are critical.

The Wasserstein metric has become increasingly important in many machine learning applications such as generative modeling, image retrieval and domain adaptation. Despite its appeal, it is often too costly to compute. This has motivated approximation methods like entropy-regularized optimal transport, downsampling, and subsampling, which trade accuracy for computational efficiency. In this paper, we consider the challenge of computing efficient approximations to the Wasserstein metric that also serve as strict upper or lower bounds. Focusing on discrete measures on regular grids, our approach involves formulating and exactly solving a Kantorovich problem on a coarse grid using a quantized measure and specially designed cost matrix, followed by an upscaling and correction stage. This is done either in the primal or dual space to obtain valid upper and lower bounds on the Wasserstein metric of the full-resolution inputs. We evaluate our methods on the DOTmark optimal transport images benchmark, demonstrating a 10x-100x speedup compared to entropy-regularized OT while keeping the approximation error below 2%.

In multivariate autoregressive models with exogenous inputs (MARX), the evolution of the signal incorporates past observations and controls, producing substantial uncertainty during parameter estimation. Bayesian inference procedures can quantify this uncertainty and propagate it towards future predictions [5], [6]. Quantified uncertainty is valuable on its own, but also useful to sensor fusion, optimal experimental design and adaptive control [7], [8], [9], [10], [11]. We present an exact recursive Bayesian estimator whose computation is distributed over a probabilistic graphical model. Bayesian inference in multivariate autoregressive models has a rich history, especially in econometrics [1], [3].

Understanding the complex interactions within dynamic multilayer networks is critical for advancements in various scientific domains. Existing models often fail to capture such networks' temporal and cross-layer dynamics. This paper introduces a novel Tensor State Space Model for Dynamic Multilayer Networks (TSSDMN), utilizing a latent space model framework. TSSDMN employs a symmetric Tucker decomposition to represent latent node features, their interaction patterns, and layer transitions. Then by fixing the latent features and allowing the interaction patterns to evolve over time, TSSDMN uniquely captures both the temporal dynamics within layers and across different layers. The model identifiability conditions are discussed. By treating latent features as variables whose posterior distributions are approximated using a mean-field variational inference approach, a variational Expectation Maximization algorithm is developed for efficient model inference. Numerical simulations and case studies demonstrate the efficacy of TSSDMN for understanding dynamic multilayer networks.

Quantifying uncertainties for machine learning (ML) models is a foundational challenge in modern data analysis. This challenge is compounded by at least two key aspects of the field: (a) inconsistent terminology surrounding uncertainty and estimation across disciplines, and (b) the varying technical requirements for establishing trustworthy uncertainties in diverse problem contexts. In this position paper, we aim to clarify the depth of these challenges by identifying these inconsistencies and articulating how different contexts impose distinct epistemic demands. We examine the current landscape of estimation targets (e.g., prediction, inference, simulation-based inference), uncertainty constructs (e.g., frequentist, Bayesian, fiducial), and the approaches used to map between them. Drawing on the literature, we highlight and explain examples of problematic mappings. To help address these issues, we advocate for standards that promote alignment between the \textit{intent} and \textit{implementation} of uncertainty quantification (UQ) approaches. We discuss several axes of trustworthiness that are necessary (if not sufficient) for reliable UQ in ML models, and show how these axes can inform the design and evaluation of uncertainty-aware ML systems. Our practical recommendations focus on scientific ML, offering illustrative cases and use scenarios, particularly in the context of simulation-based inference (SBI).

Interpretability is the next pivotal frontier in machine learning research. In the pursuit of glass box models - as opposed to black box models, like random forests or neural networks - rule induction algorithms are a logical and promising avenue, as the rules can easily be understood by humans. In our previous work, we introduced FRRI, a novel rule induction algorithm based on fuzzy rough set theory. We demonstrated experimentally that FRRI outperformed other rule induction methods with regards to accuracy and number of rules. FRRI leverages a fuzzy indiscernibility relation to partition the data space into fuzzy granules, which are then combined into a minimal covering set of rules. This indiscernibility relation is constructed by removing attributes from rules in a greedy way. This raises the question: does the order of the attributes matter? In this paper, we show that optimising only the order of attributes using known methods from fuzzy rough set theory and classical machine learning does not improve the performance of FRRI on multiple metrics. However, removing a small number of attributes using fuzzy rough feature selection during this step positively affects balanced accuracy and the average rule length.