Estimation of the conditional independence graph (CIG) of high-dimensional multivariate Gaussian time series from multi-attribute data is considered. Existing methods for graph estimation for such data are based on single-attribute models where one associates a scalar time series with each node. In multi-attribute graphical models, each node represents a random vector or vector time series. In this paper we provide a unified theoretical analysis of multi-attribute graph learning for dependent time series using a penalized log-likelihood objective function formulated in the frequency domain using the discrete Fourier transform of the time-domain data. We consider both convex (sparse-group lasso) and non-convex (log-sum and SCAD group penalties) penalty/regularization functions. We establish sufficient conditions in a high-dimensional setting for consistency (convergence of the inverse power spectral density to true value in the Frobenius norm), local convexity when using non-convex penalties, and graph recovery. We do not impose any incoherence or irrepresentability condition for our convergence results. We also empirically investigate selection of the tuning parameters based on the Bayesian information criterion, and illustrate our approach using numerical examples utilizing both synthetic and real data.

Statistically correcting measured cross sections for detector effects is an important step across many applications. In particle physics, this inverse problem is known as \textit{unfolding}. In cases with complex instruments, the distortions they introduce are often known only implicitly through simulations of the detector. Modern machine learning has enabled efficient simulation-based approaches for unfolding high-dimensional data. Among these, one of the first methods successfully deployed on experimental data is the \textsc{OmniFold} algorithm, a classifier-based Expectation-Maximization procedure. In practice, however, the forward model is only approximately specified, and the corresponding uncertainty is encoded through nuisance parameters. Building on the well-studied \textsc{OmniFold} algorithm, we show how to extend machine learning-based unfolding to incorporate nuisance parameters. Our new algorithm, called Profile \textsc{OmniFold}, is demonstrated using a Gaussian example as well as a particle physics case study using simulated data from the CMS Experiment at the Large Hadron Collider.

Inverse reinforcement learning aims to infer the reward function that explains expert behavior observed through trajectories of state--action pairs. A long-standing difficulty in classical IRL is the non-uniqueness of the recovered reward: many reward functions can induce the same optimal policy, rendering the inverse problem ill-posed. In this paper, we develop a statistical framework for Inverse Entropy-regularized Reinforcement Learning that resolves this ambiguity by combining entropy regularization with a least-squares reconstruction of the reward from the soft Bellman residual. This combination yields a unique and well-defined so-called least-squares reward consistent with the expert policy. We model the expert demonstrations as a Markov chain with the invariant distribution defined by an unknown expert policy $π^\star$ and estimate the policy by a penalized maximum-likelihood procedure over a class of conditional distributions on the action space. We establish high-probability bounds for the excess Kullback--Leibler divergence between the estimated policy and the expert policy, accounting for statistical complexity through covering numbers of the policy class. These results lead to non-asymptotic minimax optimal convergence rates for the least-squares reward function, revealing the interplay between smoothing (entropy regularization), model complexity, and sample size. Our analysis bridges the gap between behavior cloning, inverse reinforcement learning, and modern statistical learning theory.

Semi-supervised learning (SSL) constructs classifiers from datasets in which only a subset of observations is labelled, a situation that naturally arises because obtaining labels often requires expert judgement or costly manual effort. This motivates methods that integrate labelled and unlabelled data within a learning framework. Most SSL approaches assume that label absence is harmless, typically treated as missing completely at random or ignored, but in practice, the missingness process can be informative, as the chances of an observation being unlabelled may depend on the ambiguity of its feature vector. In such cases, the missingness indicators themselves provide additional information that, if properly modelled, may improve estimation efficiency. The \textbf{SSLfmm} package for R is designed to capture this behaviour by estimating the Bayes' classifier under a finite mixture model in which each component corresponding to a class follows a multivariate normal distribution. It incorporates a mixed-missingness mechanism that combines a missing completely at random (MCAR) component with a (non-ignorable) missing at random (MAR) component, the latter modelling the probability of label missingness as a logistic function of the entropy based on the features. Parameters are estimated via an Expectation--Conditional Maximisation algorithm. In the two-class Gaussian setting with arbitrary covariance matrices, the resulting classifier trained on partially labelled data may, in some cases, achieve a lower misclassification rate than the supervised version in the case where all the labels are known. The package includes a practical tool for modelling and illustrates its performance through simulated examples.

This Inaugural Lecture was given at Royal Holloway University of London in 1996. It covers an introduction to machine learning and describes various theoretical advances and practical projects in the field. The Lecture here is presented in its original format, but a few remarks have been added in 2025 to reflect recent developments, and the list of references has been updated to enhance the convenience and accuracy for readers. When did machine learning start? Maybe a good starting point is 1949, when Claude Shannon proposed a learning algorithm for chess-playing programs. Or maybe we should go back to the 1930s when Ronald Fisher developed discriminant analysis - a type of learning where the problem is to construct a decision rule that separates two types of vectors. Or could it be the 18th century when David Hume discussed the idea of induction? Or the 14th century, when William of Ockham formulated the principle of "simplicity" known as "Ockham's razor" (Ockham, by the way, is a small village not far from Royal Holloway). Or it may be that, like almost everything else in Western civilisation and culture, the origin of these ideas lies in the Mediterranean. After all, it was Aristotle who said that "we learn some things only by doing things". The field of machine learning has been greatly influenced by other disciplines and the subject is in itself not a very homogeneous discipline, but includes separate, overlapping subfields. There are many parallel lines of research in ML: inductive learning, neural networks, clustering, and theories of learning. They are all part of the more general field of machine learning.

We revisit the classic ski rental problem through the lens of Bayesian decision-making and machine-learned predictions. While traditional algorithms minimize worst-case cost without assumptions, and recent learning-augmented approaches leverage noisy forecasts with robustness guarantees, our work unifies these perspectives. We propose a discrete Bayesian framework that maintains exact posterior distributions over the time horizon, enabling principled uncertainty quantification and seamless incorporation of expert priors. Our algorithm achieves prior-dependent competitive guarantees and gracefully interpolates between worst-case and fully-informed settings. Our extensive experimental evaluation demonstrates superior empirical performance across diverse scenarios, achieving near-optimal results under accurate priors while maintaining robust worst-case guarantees. This framework naturally extends to incorporate multiple predictions, non-uniform priors, and contextual information, highlighting the practical advantages of Bayesian reasoning in online decision problems with imperfect predictions.

Simultaneous localization and mapping (SLAM) plays a fundamental role in extended reality (XR) applications. As the standards for immersion in XR continue to increase, the demands for SLAM benchmarking have become more stringent. Trajectory accuracy is the key metric, and marker-based optical motion capture (MoCap) systems are widely used to generate ground truth (GT) because of their drift-free and relatively accurate measurements. However, the precision of MoCap-based GT is limited by two factors: the spatiotemporal calibration with the device under test (DUT) and the inherent jitter in the MoCap measurements. These limitations hinder accurate SLAM benchmarking, particularly for key metrics like rotation error and inter-frame jitter, which are critical for immersive XR experiences. This paper presents a novel continuous-time maximum likelihood estimator to address these challenges. The proposed method integrates auxiliary inertial measurement unit (IMU) data to compensate for MoCap jitter. Additionally, a variable time synchronization method and a pose residual based on screw congruence constraints are proposed, enabling precise spatiotemporal calibration across multiple sensors and the DUT. Experimental results demonstrate that our approach outperforms existing methods, achieving the precision necessary for comprehensive benchmarking of state-of-the-art SLAM algorithms in XR applications. Furthermore, we thoroughly validate the practicality of our method by benchmarking several leading XR devices and open-source SLAM algorithms. The code is publicly available at https://github.com/ylab-xrpg/xr-hpgt.

Detecting regime shifts in chaotic time series is difficult because observation-space signals are entangled with intrinsic variability. We propose Parameter-Space Changepoint Detection (Param-CPD), a two-stage framework that first amortizes Bayesian inference of governing parameters with a neural posterior estimator trained by simulation-based inference, and then applies a standard CPD algorithm to the resulting parameter trajectory. In Lorenz-63 with piecewise-constant parameters, Param-CPD improves F1, reduces localization error, and reduces false positives compared to baselines of observation-space. We further verify identifiability and calibration of the inferred posteriors on stationary trajectories, explaining why parameter space offers a cleaner detection signal. Robustness analyzes of tolerance, window length, and noise indicate consistent gains. Our results show that operating in a physically interpretable parameter space enables accurate and interpretable changepoint detection in nonlinear dynamical systems.

Optimal experimental design is a classic topic in statistics, with many well-studied problems, applications, and solutions. The design problem we study is the placement of sensors to monitor spatiotemporal processes, explicitly accounting for the temporal dimension in our modeling and optimization. We observe that recent advancements in computational sciences often yield large datasets based on physics-based simulations, which are rarely leveraged in experimental design. We introduce a novel model-based sensor placement criterion, along with a highly-efficient optimization algorithm, which integrates physics-based simulations and Bayesian experimental design principles to identify sensor networks that "minimize information loss" from simulated data. Our technique relies on sparse variational inference and (separable) Gauss-Markov priors, and thus may adapt many techniques from Bayesian experimental design. We validate our method through a case study monitoring air temperature in Phoenix, Arizona, using state-of-the-art physics-based simulations. Our results show our framework to be superior to random or quasi-random sampling, particularly with a limited number of sensors. We conclude by discussing practical considerations and implications of our framework, including more complex modeling tools and real-world deployments.

A scalable Bayesian machine learning framework is introduced for estimating scalar properties of an unknown quantum state from measurement data, which bypasses full density matrix reconstruction. This work is the first to integrate the classical shadows protocol with a permutation-invariant set transformer architecture, enabling the approach to predict and correct bias in existing estimators to approximate the true Bayesian posterior mean. Measurement outcomes are encoded as fixed-dimensional feature vectors, and the network outputs a residual correction to a baseline estimator. Scalability to large quantum systems is ensured by the polynomial dependence of input size on system size and number of measurements. On Greenberger-Horne-Zeilinger state fidelity and second-order Rényi entropy estimation tasks -- using random Pauli and random Clifford measurements -- this Bayesian estimator always achieves lower mean squared error than classical shadows alone, with more than a 99\% reduction in the few copy regime.