Simulation of conditioned diffusion processes is an essential tool in inference for stochastic processes, data imputation, generative modelling, and geometric statistics. Whilst simulating diffusion bridge processes is already difficult on Euclidean spaces, when considering diffusion processes on Riemannian manifolds the geometry brings in further complications. In even higher generality, advancing from Riemannian to sub-Riemannian geometries introduces hypoellipticity, and the possibility of finding appropriate explicit approximations for the score of the diffusion process is removed. We handle these challenges and construct a method for bridge simulation on sub-Riemannian manifolds by demonstrating how recent progress in machine learning can be modified to allow for training of score approximators on sub-Riemannian manifolds. Since gradients dependent on the horizontal distribution, we generalise the usual notion of denoising loss to work with non-holonomic frames using a stochastic Taylor expansion, and we demonstrate the resulting scheme both explicitly on the Heisenberg group and more generally using adapted coordinates. We perform numerical experiments exemplifying samples from the bridge process on the Heisenberg group and the concentration of this process for small time.

The idea of the inheritance of biological processes, such as the developmental process or the life cycle of an organism, has been discussed in the biology literature, but formal mathematical descriptions and plausible data analysis frameworks are lacking. We introduce an extension of the nested Dirichlet Process (nDP) to a multiscale model to aid in understanding the mechanisms by which biological processes are inherited, remain stable, and are modified across generations. To address these issues, we introduce Nested Inheritance Dynamics Algorithm (NIDA). At its primary level, NIDA encompasses all processes unfolding within an individual organism's lifespan. The secondary level delineates the dynamics through which these processes evolve or remain stable over time. This framework allows for the specification of a physical system model at either scale, thus promoting seamless integration with established models of development and heredity.

As a key part of modern online platforms, online decision-making plays a crucial role in a variety of settings, particularly related to the Internet. Two landmark examples that have been widely studied are dynamic pricing and online reviews. Online review systems constitute powerful platforms for users to get informed about the product and for the firm to understand how a given market is receiving the product. The study of these systems has been vast for the last two decades [6, 10], and more recently, modeling simple like/dislike reviews as bandits problems have become standard [1, 2, 3, 13, 16, 18]. Dynamic pricing, on the other hand, is an active area of research in economics, computer science, and operations research [12, 14], and has become a common practice in several industries such as transportation and retail. There has been a growing interest in combining the two areas as a way to design more effective pricing mechanisms that gather information from current reviews to update prices and make the product more attractive [5, 11, 17]. In particular, [5] considers social learning with non-Bayesian agents in a market with like & dislike reviews, and the resulting pricing decision of a monopolist.

Graduate School of Information Science and Technology Osaka University Osaka, Japan thanasutives@ai.sanken.osaka-u.ac.jp Ken-ichi Fukui SANKEN (The Institute of Scientific and Industrial Research) Osaka University Osaka, Japan fukui@ai.sanken.osaka-u.ac.jp The uncertainty-penalized information criterion (UBIC) has been proposed as a new model-selection criterion for data-driven partial differential equation (PDE) discovery. In this paper, we show that using the UBIC is equivalent to employing the conventional BIC to a set of overparameterized models derived from the potential regression models of different complexity measures. The result indicates that the asymptotic property of the UBIC and BIC holds indifferently.

In engineering, accurately modeling nonlinear dynamic systems from data contaminated by noise is both essential and complex. Established Sequential Monte Carlo (SMC) methods, used for the Bayesian identification of these systems, facilitate the quantification of uncertainty in the parameter identification process. A significant challenge in this context is the numerical integration of continuous-time ordinary differential equations (ODEs), crucial for aligning theoretical models with discretely sampled data. This integration introduces additional numerical uncertainty, a factor that is often over looked. To address this issue, the field of probabilistic numerics combines numerical methods, such as numerical integration, with probabilistic modeling to offer a more comprehensive analysis of total uncertainty. By retaining the accuracy of classical deterministic methods, these probabilistic approaches offer a deeper understanding of the uncertainty inherent in the inference process. This paper demonstrates the application of a probabilistic numerical method for solving ODEs in the joint parameter-state identification of nonlinear dynamic systems. The presented approach efficiently identifies latent states and system parameters from noisy measurements. Simultaneously incorporating probabilistic solutions to the ODE in the identification challenge. The methodology's primary advantage lies in its capability to produce posterior distributions over system parameters, thereby representing the inherent uncertainties in both the data and the identification process.

Surrogate models provide a quick-to-evaluate approximation to complex computational models and are essential for multi-query problems like design optimisation. The inputs of current computational models are usually high-dimensional and uncertain. We consider Bayesian inference for constructing statistical surrogates with input uncertainties and intrinsic dimensionality reduction. The surrogates are trained by fitting to data from prevalent deterministic computational models. The assumed prior probability density of the surrogate is a Gaussian process. We determine the respective posterior probability density and parameters of the posited statistical model using variational Bayes. The non-Gaussian posterior is approximated by a simpler trial density with free variational parameters and the discrepancy between them is measured using the Kullback-Leibler (KL) divergence. We employ the stochastic gradient method to compute the variational parameters and other statistical model parameters by minimising the KL divergence. We demonstrate the accuracy and versatility of the proposed reduced dimension variational Gaussian process (RDVGP) surrogate on illustrative and robust structural optimisation problems with cost functions depending on a weighted sum of the mean and standard deviation of model outputs.

In this paper, we explore the optimization of metal recycling with a focus on real-time differentiation between alloys of copper and aluminium. Spectral data, obtained through Prompt Gamma Neutron Activation Analysis (PGNAA), is utilized for classification. The study compares data from two detectors, cerium bromide (CeBr$_{3}$) and high purity germanium (HPGe), considering their energy resolution and sensitivity. We test various data generation, preprocessing, and classification methods, with Maximum Likelihood Classifier (MLC) and Conditional Variational Autoencoder (CVAE) yielding the best results. The study also highlights the impact of different detector types on classification accuracy, with CeBr$_{3}$ excelling in short measurement times and HPGe performing better in longer durations. The findings suggest the importance of selecting the appropriate detector and methodology based on specific application requirements.

Graphical models, including Graph Neural Networks (GNNs) and Probabilistic Graphical Models (PGMs), have demonstrated their exceptional capabilities across numerous fields. These models necessitate effective uncertainty quantification to ensure reliable decision-making amid the challenges posed by model training discrepancies and unpredictable testing scenarios. This survey examines recent works that address uncertainty quantification within the model architectures, training, and inference of GNNs and PGMs. We aim to provide an overview of the current landscape of uncertainty in graphical models by organizing the recent methods into uncertainty representation and handling. By summarizing state-of-the-art methods, this survey seeks to deepen the understanding of uncertainty quantification in graphical models, thereby increasing their effectiveness and safety in critical applications.

We consider a variant of online binary classification where a learner sequentially assigns labels ($0$ or $1$) to items with unknown true class. If, but only if, the learner chooses label $1$ they immediately observe the true label of the item. The learner faces a trade-off between short-term classification accuracy and long-term information gain. This problem has previously been studied under the name of the `apple tasting' problem. We revisit this problem as a partial monitoring problem with side information, and focus on the case where item features are linked to true classes via a logistic regression model. Our principal contribution is a study of the performance of Thompson Sampling (TS) for this problem. Using recently developed information-theoretic tools, we show that TS achieves a Bayesian regret bound of an improved order to previous approaches. Further, we experimentally verify that efficient approximations to TS and Information Directed Sampling via P\'{o}lya-Gamma augmentation have superior empirical performance to existing methods.

Formulating dynamical models for physical phenomena is essential for understanding the interplay between the different mechanisms and predicting the evolution of physical states. However, a dynamical model alone is often insufficient to address these fundamental tasks, as it suffers from model errors and uncertainties. One common remedy is to rely on data assimilation, where the state estimate is updated with observations of the true system. Ensemble filters sequentially assimilate observations by updating a set of samples over time. They operate in two steps: a forecast step that propagates each sample through the dynamical model and an analysis step that updates the samples with incoming observations. For accurate and robust predictions of dynamical systems, discrete solutions must preserve their critical invariants. While modern numerical solvers satisfy these invariants, existing invariant-preserving analysis steps are limited to Gaussian settings and are often not compatible with classical regularization techniques of ensemble filters, e.g., inflation and covariance tapering. The present work focuses on preserving linear invariants, such as mass, stoichiometric balance of chemical species, and electrical charges. Using tools from measure transport theory (Spantini et al., 2022, SIAM Review), we introduce a generic class of nonlinear ensemble filters that automatically preserve desired linear invariants in non-Gaussian filtering problems. By specializing this framework to the Gaussian setting, we recover a constrained formulation of the Kalman filter. Then, we show how to combine existing regularization techniques for the ensemble Kalman filter (Evensen, 1994, J. Geophys. Res.) with the preservation of the linear invariants. Finally, we assess the benefits of preserving linear invariants for the ensemble Kalman filter and nonlinear ensemble filters.