Missing Not at Random (MNAR) and nonnormal data are challenging to handle. Traditional missing data analytical techniques such as full information maximum likelihood estimation (FIML) may fail with nonnormal data as they are built on normal distribution assumptions. Two-Stage Robust Estimation (TSRE) does manage nonnormal data, but both FIML and TSRE are less explored in longitudinal studies under MNAR conditions with nonnormal distributions. Unlike traditional statistical approaches, machine learning approaches do not require distributional assumptions about the data. More importantly, they have shown promise for MNAR data; however, their application in longitudinal studies, addressing both Missing at Random (MAR) and MNAR scenarios, is also underexplored. This study utilizes Monte Carlo simulations to assess and compare the effectiveness of six analytical techniques for missing data within the growth curve modeling framework. These techniques include traditional approaches like FIML and TSRE, machine learning approaches by single imputation (K-Nearest Neighbors and missForest), and machine learning approaches by multiple imputation (micecart and miceForest). We investigate the influence of sample size, missing data rate, missing data mechanism, and data distribution on the accuracy and efficiency of model estimation. Our findings indicate that FIML is most effective for MNAR data among the tested approaches. TSRE excels in handling MAR data, while missForest is only advantageous in limited conditions with a combination of very skewed distributions, very large sample sizes (e.g., n larger than 1000), and low missing data rates.

Recently proposed quasi-Bayesian (QB) methods initiated a new era in Bayesian computation by directly constructing the Bayesian predictive distribution through recursion, removing the need for expensive computations involved in sampling the Bayesian posterior distribution. This has proved to be data-efficient for univariate predictions, but extensions to multiple dimensions rely on a conditional decomposition resulting from predefined assumptions on the kernel of the Dirichlet Process Mixture Model, which is the implicit nonparametric model used. Here, we propose a different way to extend Quasi-Bayesian prediction to high dimensions through the use of Sklar's theorem by decomposing the predictive distribution into one-dimensional predictive marginals and a high-dimensional copula. Thus, we use the efficient recursive QB construction for the one-dimensional marginals and model the dependence using highly expressive vine copulas. Further, we tune hyperparameters using robust divergences (eg. energy score) and show that our proposed Quasi-Bayesian Vine (QB-Vine) is a fully non-parametric density estimator with \emph{an analytical form} and convergence rate independent of the dimension of data in some situations. Our experiments illustrate that the QB-Vine is appropriate for high dimensional distributions ($\sim$64), needs very few samples to train ($\sim$200) and outperforms state-of-the-art methods with analytical forms for density estimation and supervised tasks by a considerable margin.

The need for regression models to predict circular values arises in many scientific fields. In this work we explore a family of expressive and interpretable distributions over circle-valued random functions related to Gaussian processes targeting two Euclidean dimensions conditioned on the unit circle. The resulting probability model has connections with continuous spin models in statistical physics. Moreover, its density is very simple and has maximum-entropy, unlike previous Gaussian process-based approaches, which use wrapping or radial marginalization. For posterior inference, we introduce a new Stratonovich-like augmentation that lends itself to fast Markov Chain Monte Carlo sampling. We argue that transductive learning in these models favors a Bayesian approach to the parameters. We present experiments applying this model to the prediction of (i) wind directions and (ii) the percentage of the running gait cycle as a function of joint angles.

Variational Inference (VI) is a commonly used technique for approximate Bayesian inference and uncertainty estimation in deep learning models, yet it comes at a computational cost, as it doubles the number of trainable parameters to represent uncertainty. This rapidly becomes challenging in high-dimensional settings and motivates the use of alternative techniques for inference, such as Monte Carlo Dropout (MCD) or Spectral-normalized Neural Gaussian Process (SNGP). However, such methods have seen little adoption in survival analysis, and VI remains the prevalent approach for training probabilistic neural networks. In this paper, we investigate how to train deep probabilistic survival models in large datasets without introducing additional overhead in model complexity. To achieve this, we adopt three probabilistic approaches, namely VI, MCD, and SNGP, and evaluate them in terms of their prediction performance, calibration performance, and model complexity. In the context of probabilistic survival analysis, we investigate whether non-VI techniques can offer comparable or possibly improved prediction performance and uncertainty calibration compared to VI. In the MIMIC-IV dataset, we find that MCD aligns with VI in terms of the concordance index (0.748 vs. 0.743) and mean absolute error (254.9 vs. 254.7) using hinge loss, while providing C-calibrated uncertainty estimates. Moreover, our SNGP implementation provides D-calibrated survival functions in all datasets compared to VI (4/4 vs. 2/4, respectively). Our work encourages the use of techniques alternative to VI for survival analysis in high-dimensional datasets, where computational efficiency and overhead are of concern.

In recent years, predictive maintenance (PMx) has gained prominence for its potential to enhance efficiency, automation, accuracy, and cost-effectiveness while reducing human involvement. Importantly, PMx has evolved in tandem with digital advancements, such as Big Data and the Internet of Things (IOT). These technological strides have enabled Artificial Intelligence (AI) to revolutionize PMx processes, with increasing capacities for real-time automation of monitoring, analysis, and prediction tasks. However, PMx still faces challenges such as poor explainability and sample inefficiency in data-driven methods and high complexity in physics-based models, hindering broader adoption. This paper posits that Digital Twins (DTs) can be integrated into PMx to overcome these challenges, paving the way for more automated PMx applications across various stakeholders. Despite their potential, current DTs have not fully matured to bridge existing gaps. Our paper provides a comprehensive roadmap for DT evolution, addressing current limitations to foster large-scale automated PMx progression. We structure our approach in three stages: First, we reference prior work where we identified and defined the Information Requirements (IRs) and Functional Requirements (FRs) for PMx, forming the blueprint for a unified framework. Second, we conduct a literature review to assess current DT applications integrating these IRs and FRs, revealing standardized DT models and tools that support automated PMx. Lastly, we highlight gaps in current DT implementations, particularly those IRs and FRs not fully supported, and outline the necessary components for a comprehensive, automated PMx system. Our paper concludes with research directions aimed at seamlessly integrating DTs into the PMx paradigm to achieve this ambitious vision.

Due to their flexibility and theoretical tractability Gaussian process (GP) regression models have become a central topic in modern statistics and machine learning. While the true posterior in these models is given explicitly, numerical evaluations depend on the inversion of the augmented kernel matrix $ K + \sigma^2 I $, which requires up to $ O(n^3) $ operations. For large sample sizes n, which are typically given in modern applications, this is computationally infeasible and necessitates the use of an approximate version of the posterior. Although such methods are widely used in practice, they typically have very limtied theoretical underpinning. In this context, we analyze a class of recently proposed approximation algorithms from the field of Probabilistic numerics. They can be interpreted in terms of Lanczos approximate eigenvectors of the kernel matrix or a conjugate gradient approximation of the posterior mean, which are particularly advantageous in truly large scale applications, as they are fundamentally only based on matrix vector multiplications amenable to the GPU acceleration of modern software frameworks. We combine result from the numerical analysis literature with state of the art concentration results for spectra of kernel matrices to obtain minimax contraction rates. Our theoretical findings are illustrated by numerical experiments.

Machine learning (ML) and Artificial Intelligence (AI) have fueled remarkable advancements, particularly in healthcare. Within medical imaging, ML models hold the promise of improving disease diagnoses, treatment planning, and post-treatment monitoring. Various computer vision tasks like image classification, object detection, and image segmentation are poised to become routine in clinical analysis. However, privacy concerns surrounding patient data hinder the assembly of large training datasets needed for developing and training accurate, robust, and generalizable models. Federated Learning (FL) emerges as a compelling solution, enabling organizations to collaborate on ML model training by sharing model training information (gradients) rather than data (e.g., medical images). FL's distributed learning framework facilitates inter-institutional collaboration while preserving patient privacy. However, FL, while robust in privacy preservation, faces several challenges. Sensitive information can still be gleaned from shared gradients that are passed on between organizations during model training. Additionally, in medical imaging, quantifying model confidence\uncertainty accurately is crucial due to the noise and artifacts present in the data. Uncertainty estimation in FL encounters unique hurdles due to data heterogeneity across organizations. This paper offers a comprehensive review of FL, privacy preservation, and uncertainty estimation, with a focus on medical imaging. Alongside a survey of current research, we identify gaps in the field and suggest future directions for FL research to enhance privacy and address noisy medical imaging data challenges.

Sum-product networks (SPNs) are probabilistic models characterized by exact and fast evaluation of fundamental probabilistic operations. Its superior computational tractability has led to applications in many fields, such as machine learning with time constraints or accuracy requirements and real-time systems. The structural constraints of SPNs supporting fast inference, however, lead to increased learning-time complexity and can be an obstacle to building highly expressive SPNs. This study aimed to develop a Bayesian learning approach that can be efficiently implemented on large-scale SPNs. We derived a new full conditional probability of Gibbs sampling by marginalizing multiple random variables to expeditiously obtain the posterior distribution. The complexity analysis revealed that our sampling algorithm works efficiently even for the largest possible SPN. Furthermore, we proposed a hyperparameter tuning method that balances the diversity of the prior distribution and optimization efficiency in large-scale SPNs. Our method has improved learning-time complexity and demonstrated computational speed tens to more than one hundred times faster and superior predictive performance in numerical experiments on more than 20 datasets.

We propose a scalable variational Bayes method for statistical inference for a single or low-dimensional subset of the coordinates of a high-dimensional parameter in sparse linear regression. Our approach relies on assigning a mean-field approximation to the nuisance coordinates and carefully modelling the conditional distribution of the target given the nuisance. This requires only a preprocessing step and preserves the computational advantages of mean-field variational Bayes, while ensuring accurate and reliable inference for the target parameter, including for uncertainty quantification. We investigate the numerical performance of our algorithm, showing that it performs competitively with existing methods. We further establish accompanying theoretical guarantees for estimation and uncertainty quantification in the form of a Bernstein--von Mises theorem.

Large Language Models (LLMs) often suffer from overconfidence during inference, particularly when adapted to downstream domain-specific tasks with limited data. Previous work addresses this issue by employing approximate Bayesian estimation after the LLMs are trained, enabling them to quantify uncertainty. However, such post-training approaches' performance is severely limited by the parameters learned during training. In this paper, we go beyond post-training Bayesianization and propose Bayesian Low-Rank Adaptation by Backpropagation (BLoB), an algorithm that continuously and jointly adjusts both the mean and covariance of LLM parameters throughout the whole fine-tuning process. Our empirical results verify the effectiveness of BLoB in terms of generalization and uncertainty estimation, when evaluated on both in-distribution and out-of-distribution data.