Several popular variational bounds involving importance weighting ideas have been proposed to generalize and improve on the Evidence Lower BOund (ELBO) in the context of maximum likelihood optimization, such as the Importance Weighted Auto-Encoder (IWAE) and the Variational R\'enyi (VR) bounds. The methodology to learn the parameters of interest using these bounds typically amounts to running gradient-based variational inference algorithms that incorporate the reparameterization trick. However, the way the choice of the variational bound impacts the outcome of variational inference algorithms can be unclear. Recently, the VR-IWAE bound was introduced as a variational bound that unifies the ELBO, IWAE and VR bounds methodologies. In this paper, we provide two analyses for the reparameterized and doubly-reparameterized gradient estimators of the VR-IWAE bound, which reveal the advantages and limitations of these gradient estimators while enabling us to compare of the ELBO, IWAE and VR bounds methodologies. Our work advances the understanding of importance weighted variational inference methods and we illustrate our theoretical findings empirically.

Bayesian Optimal Experimental Design (BOED) is a powerful tool to reduce the cost of running a sequence of experiments. When based on the Expected Information Gain (EIG), design optimization corresponds to the maximization of some intractable expected {\it contrast} between prior and posterior distributions. Scaling this maximization to high dimensional and complex settings has been an issue due to BOED inherent computational complexity. In this work, we introduce an {\it expected posterior} distribution with cost-effective sampling properties and provide a tractable access to the EIG contrast maximization via a new EIG gradient expression. Diffusion-based samplers are used to compute the dynamics of the expected posterior and ideas from bi-level optimization are leveraged to derive an efficient joint sampling-optimization loop, without resorting to lower bound approximations of the EIG. The resulting efficiency gain allows to extend BOED to the well-tested generative capabilities of diffusion models. By incorporating generative models into the BOED framework, we expand its scope and its use in scenarios that were previously impractical. Numerical experiments and comparison with state-of-the-art methods show the potential of the approach.

Differentiable programming allows for derivatives of functions implemented via computer code to be calculated automatically. These derivatives are calculated using automatic differentiation (AD). This thesis explores two applications of differentiable programming to computational plasma physics. First, we consider how differentiable programming can be used to simplify and improve stellarator optimization. We introduce a stellarator coil design code (FOCUSADD) that uses gradient-based optimization to produce stellarator coils with finite build. Because we use reverse mode AD, which can compute gradients of scalar functions with the same computational complexity as the function, FOCUSADD is simple, flexible, and efficient. We then discuss two additional applications of AD in stellarator optimization. Second, we explore how machine learning (ML) can be used to improve or replace the numerical methods used to solve partial differential equations (PDEs), focusing on time-dependent PDEs in fluid mechanics relevant to plasma physics. Differentiable programming allows neural networks and other techniques from ML to be embedded within numerical methods. This is a promising, but relatively new, research area. We focus on two basic questions. First, can we design ML-based PDE solvers that have the same guarantees of conservation, stability, and positivity that standard numerical methods do? The answer is yes; we introduce error-correcting algorithms that preserve invariants of time-dependent PDEs. Second, which types of ML-based solvers work best at solving PDEs? We perform a systematic review of the scientific literature on solving PDEs with ML. Unfortunately we discover two issues, weak baselines and reporting biases, that affect the interpretation reproducibility of a significant majority of published research. We conclude that using ML to solve PDEs is not as promising as we initially believed.

Monitoring networks increasingly aim to assimilate data from a large number of diverse sensors covering many sensing modalities. Bayesian optimal experimental design (OED) seeks to identify data, sensor configurations, or experiments which can optimally reduce uncertainty and hence increase the performance of a monitoring network. Information theory guides OED by formulating the choice of experiment or sensor placement as an optimization problem that maximizes the expected information gain (EIG) about quantities of interest given prior knowledge and models of expected observation data. Therefore, within the context of seismo-acoustic monitoring, we can use Bayesian OED to configure sensor networks by choosing sensor locations, types, and fidelity in order to improve our ability to identify and locate seismic sources. In this work, we develop the framework necessary to use Bayesian OED to optimize a sensor network's ability to locate seismic events from arrival time data of detected seismic phases at the regional-scale. Bayesian OED requires four elements: 1) A likelihood function that describes the distribution of detection and travel time data from the sensor network, 2) A Bayesian solver that uses a prior and likelihood to identify the posterior distribution of seismic events given the data, 3) An algorithm to compute EIG about seismic events over a dataset of hypothetical prior events, 4) An optimizer that finds a sensor network which maximizes EIG. Once we have developed this framework, we explore many relevant questions to monitoring such as: how to trade off sensor fidelity and earth model uncertainty; how sensor types, number, and locations influence uncertainty; and how prior models and constraints influence sensor placement.

We propose a scalable kinetic Langevin dynamics algorithm for sampling parameter spaces of big data and AI applications. Our scheme combines a symmetric forward/backward sweep over minibatches with a symmetric discretization of Langevin dynamics. For a particular Langevin splitting method (UBU), we show that the resulting Symmetric Minibatch Splitting-UBU (SMS-UBU) integrator has bias $O(h^2 d^{1/2})$ in dimension $d>0$ with stepsize $h>0$, despite only using one minibatch per iteration, thus providing excellent control of the sampling bias as a function of the stepsize. We apply the algorithm to explore local modes of the posterior distribution of Bayesian neural networks (BNNs) and evaluate the calibration performance of the posterior predictive probabilities for neural networks with convolutional neural network architectures for classification problems on three different datasets (Fashion-MNIST, Celeb-A and chest X-ray). Our results indicate that BNNs sampled with SMS-UBU can offer significantly better calibration performance compared to standard methods of training and stochastic weight averaging.

Reasoning is key to many decision making processes. It requires consolidating a set of rule-like premises that are often associated with degrees of uncertainty and observations to draw conclusions. In this work, we address both the case where premises are specified as numeric probabilistic rules and situations in which humans state their estimates using words expressing degrees of certainty. Existing probabilistic reasoning datasets simplify the task, e.g., by requiring the model to only rank textual alternatives, by including only binary random variables, or by making use of a limited set of templates that result in less varied text. In this work, we present QUITE, a question answering dataset of real-world Bayesian reasoning scenarios with categorical random variables and complex relationships. QUITE provides high-quality natural language verbalizations of premises together with evidence statements and expects the answer to a question in the form of an estimated probability. We conduct an extensive set of experiments, finding that logic-based models outperform out-of-the-box large language models on all reasoning types (causal, evidential, and explaining-away). Our results provide evidence that neuro-symbolic models are a promising direction for improving complex reasoning. We release QUITE and code for training and experiments on Github.

In this study, we propose a robust set of features derived from a thorough research of contemporary practices in voice pathology detection. The feature set is based on the combination of acoustic handcrafted features. Additionally, we introduce pitch difference as a novel feature. We combine this feature set, containing data from the publicly available Saarbr\"ucken Voice Database (SVD), with preprocessing using the K-Means Synthetic Minority Over-Sampling Technique algorithm to address class imbalance. Moreover, we applied multiple ML models as binary classifiers. We utilized support vector machine, k-nearest neighbors, naive Bayes, decision tree, random forest and AdaBoost classifiers. To determine the best classification approach, we performed grid search on feasible hyperparameters of respective classifiers and subsections of features. Our approach has achieved the state-of-the-art performance, measured by unweighted average recall in voice pathology detection on SVD database. We intentionally omit accuracy as it is highly biased metric in case of unbalanced data compared to aforementioned metrics. The results are further enhanced by eliminating the potential overestimation of the results with repeated stratified cross-validation. This advancement demonstrates significant potential for the clinical deployment of ML methods, offering a valuable tool for an objective examination of voice pathologies. To support our claims, we provide a publicly available GitHub repository with DOI 10.5281/zenodo.13771573. Finally, we provide REFORMS checklist.

Bayesian optimisation for real-world problems is often performed interactively with human experts, and integrating their domain knowledge is key to accelerate the optimisation process. We consider a setup where experts provide advice on the next query point through binary accept/reject recommendations (labels). Experts' labels are often costly, requiring efficient use of their efforts, and can at the same time be unreliable, requiring careful adjustment of the degree to which any expert is trusted. We introduce the first principled approach that provides two key guarantees. (1) Handover guarantee: similar to a no-regret property, we establish a sublinear bound on the cumulative number of experts' binary labels. Initially, multiple labels per query are needed, but the number of expert labels required asymptotically converges to zero, saving both expert effort and computation time. (2) No-harm guarantee with data-driven trust level adjustment: our adaptive trust level ensures that the convergence rate will not be worse than the one without using advice, even if the advice from experts is adversarial. Unlike existing methods that employ a user-defined function that hand-tunes the trust level adjustment, our approach enables data-driven adjustments. Real-world applications empirically demonstrate that our method not only outperforms existing baselines, but also maintains robustness despite varying labelling accuracy, in tasks of battery design with human experts.

Reliable estimation of predictive uncertainty is crucial for machine learning applications, particularly in high-stakes scenarios where hedging against risks is essential. Despite its significance, a consensus on the correct measurement of predictive uncertainty remains elusive. In this work, we return to first principles to develop a fundamental framework of information-theoretic predictive uncertainty measures. Our proposed framework categorizes predictive uncertainty measures according to two factors: (I) The predicting model (II) The approximation of the true predictive distribution. Examining all possible combinations of these two factors, we derive a set of predictive uncertainty measures that includes both known and newly introduced ones. We empirically evaluate these measures in typical uncertainty estimation settings, such as misclassification detection, selective prediction, and out-of-distribution detection. The results show that no single measure is universal, but the effectiveness depends on the specific setting. Thus, our work provides clarity about the suitability of predictive uncertainty measures by clarifying their implicit assumptions and relationships.

The aim of the notes is to demonstrate the potential for ideas in machine learning to impact on the fields of inverse problems and data assimilation. The perspective is one that is primarily aimed at researchers from inverse problems and/or data assimilation who wish to see a mathematical presentation of machine learning as it pertains to their fields. As a by-product of the presentation we present a succinct mathematical treatment of various topics in machine learning. The material on machine learning, along with some other related topics, is summarized in Part III, Appendix. Part I of the notes is concerned with inverse problems, employing material from Part III; Part II of the notes is concerned with data assimilation, employing material from Parts I and III.