Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs), also known as ODE filters, have been established as efficient methods for quantifying numerical uncertainty in the solution of ODEs. In practical applications, however, the underlying dynamical system often contains uncertain parameters, requiring the propagation of this model uncertainty to the ODE solution. In this paper, we demonstrate that ODE filters, despite their probabilistic nature, do not automatically solve this uncertainty propagation problem. To address this limitation, we present a novel approach that combines ODE filters with numerical quadrature to properly marginalize over uncertain parameters, while accounting for both parameter uncertainty and numerical solver uncertainty. Experiments across multiple dynamical systems demonstrate that the resulting uncertainty estimates closely match reference solutions. Notably, we show how the numerical uncertainty from the ODE solver can help prevent overconfidence in the propagated uncertainty estimates, especially when using larger step sizes. Our results illustrate that probabilistic numerical methods can effectively quantify both numerical and parametric uncertainty in dynamical systems.

We introduce Dirichlet Process Posterior Sampling (DPPS), a Bayesian non-parametric algorithm for multi-arm bandits based on Dirichlet Process (DP) priors. Like Thompson-sampling, DPPS is a probability-matching algorithm, i.e., it plays an arm based on its posterior-probability of being optimal. Instead of assuming a parametric class for the reward generating distribution of each arm, and then putting a prior on the parameters, in DPPS the reward generating distribution is directly modeled using DP priors. DPPS provides a principled approach to incorporate prior belief about the bandit environment, and in the noninformative limit of the DP posteriors (i.e. Bayesian Bootstrap), we recover Non Parametric Thompson Sampling (NPTS), a popular non-parametric bandit algorithm, as a special case of DPPS. We employ stick-breaking representation of the DP priors, and show excellent empirical performance of DPPS in challenging synthetic and real world bandit environments. Finally, using an information-theoretic analysis, we show non-asymptotic optimality of DPPS in the Bayesian regret setup.

Research in adversarial machine learning (AML) has shown that statistical models are vulnerable to maliciously altered data. However, despite advances in Bayesian machine learning models, most AML research remains concentrated on classical techniques. Therefore, we focus on extending the white-box model poisoning paradigm to attack generic Bayesian inference, highlighting its vulnerability in adversarial contexts. A suite of attacks are developed that allow an attacker to steer the Bayesian posterior toward a target distribution through the strategic deletion and replication of true observations, even when only sampling access to the posterior is available. Analytic properties of these algorithms are proven and their performance is empirically examined in both synthetic and real-world scenarios. With relatively little effort, the attacker is able to substantively alter the Bayesian's beliefs and, by accepting more risk, they can mold these beliefs to their will. By carefully constructing the adversarial posterior, surgical poisoning is achieved such that only targeted inferences are corrupted and others are minimally disturbed.

We propose a novel approach for learning causal response representations. Our method aims to extract directions in which a multidimensional outcome is most directly caused by a treatment variable. By bridging conditional independence testing with causal representation learning, we formulate an optimisation problem that maximises the evidence against conditional independence between the treatment and outcome, given a conditioning set. This formulation employs flexible regression models tailored to specific applications, creating a versatile framework. The problem is addressed through a generalised eigenvalue decomposition. We show that, under mild assumptions, the distribution of the largest eigenvalue can be bounded by a known $F$-distribution, enabling testable conditional independence. We also provide theoretical guarantees for the optimality of the learned representation in terms of signal-to-noise ratio and Fisher information maximisation. Finally, we demonstrate the empirical effectiveness of our approach in simulation and real-world experiments. Our results underscore the utility of this framework in uncovering direct causal effects within complex, multivariate settings.

Gaussian processes (GPs) are powerful models for human-in-the-loop experiments due to their flexibility and well-calibrated uncertainty. However, GPs modeling human responses typically ignore auxiliary information, including a priori domain expertise and non-task performance information like user confidence ratings. We propose mixed likelihood variational GPs to leverage auxiliary information, which combine multiple likelihoods in a single evidence lower bound to model multiple types of data. We demonstrate the benefits of mixing likelihoods in three real-world experiments with human participants. First, we use mixed likelihood training to impose prior knowledge constraints in GP classifiers, which accelerates active learning in a visual perception task where users are asked to identify geometric errors resulting from camera position errors in virtual reality. Second, we show that leveraging Likert scale confidence ratings by mixed likelihood training improves model fitting for haptic perception of surface roughness. Lastly, we show that Likert scale confidence ratings improve human preference learning in robot gait optimization. The modeling performance improvements found using our framework across this diverse set of applications illustrates the benefits of incorporating auxiliary information into active learning and preference learning by using mixed likelihoods to jointly model multiple inputs.

A generative modeling framework is proposed that combines diffusion models and manifold learning to efficiently sample data densities on manifolds. The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent) spaces in the high-dimensional data (ambient) space. Two approaches for sampling from the latent data density are described. The first is a score-based diffusion model, which is trained to map a standard normal distribution to the latent data distribution using a neural network. The second one involves solving an It\^o stochastic differential equation in the latent space. Additional realizations of the data are generated by lifting the samples back to the ambient space using Double Diffusion Maps, a recently introduced technique typically employed in studying dynamical system reduction; here the focus lies in sampling densities rather than system dynamics. The proposed approaches enable sampling high dimensional data densities restricted to low-dimensional, a priori unknown manifolds. The efficacy of the proposed framework is demonstrated through a benchmark problem and a material with multiscale structure.

Bayesian filtering for high-dimensional nonlinear stochastic dynamical systems is a fundamental yet challenging problem in many fields of science and engineering. Existing methods face significant obstacles: Gaussian-based filters struggle with non-Gaussian distributions, while sequential Monte Carlo methods are computationally intensive and prone to particle degeneracy in high dimensions. Although generative models in machine learning have made significant progress in modeling high-dimensional non-Gaussian distributions, their inefficiency in online updating limits their applicability to filtering problems. To address these challenges, we propose a flow-based Bayesian filter (FBF) that integrates normalizing flows to construct a novel latent linear state-space model with Gaussian filtering distributions. This framework facilitates efficient density estimation and sampling using invertible transformations provided by normalizing flows, and it enables the construction of filters in a data-driven manner, without requiring prior knowledge of system dynamics or observation models. Numerical experiments demonstrate the superior accuracy and efficiency of FBF.

Mixture of experts (MoE) has recently emerged as an effective framework to advance the efficiency and scalability of machine learning models by softly dividing complex tasks among multiple specialized sub-models termed experts. Central to the success of MoE is an adaptive softmax gating mechanism which takes responsibility for determining the relevance of each expert to a given input and then dynamically assigning experts their respective weights. Despite its widespread use in practice, a comprehensive study on the effects of the softmax gating on the MoE has been lacking in the literature. To bridge this gap in this paper, we perform a convergence analysis of parameter estimation and expert estimation under the MoE equipped with the standard softmax gating or its variants, including a dense-to-sparse gating and a hierarchical softmax gating, respectively. Furthermore, our theories also provide useful insights into the design of sample-efficient expert structures. In particular, we demonstrate that it requires polynomially many data points to estimate experts satisfying our proposed \emph{strong identifiability} condition, namely a commonly used two-layer feed-forward network. In stark contrast, estimating linear experts, which violate the strong identifiability condition, necessitates exponentially many data points as a result of intrinsic parameter interactions expressed in the language of partial differential equations. All the theoretical results are substantiated with a rigorous guarantee.

Since its origin in the thermodynamics of the 19th century, the concept of entropy has also permeated other fields of physics and mathematics, such as Classical and Quantum Statistical Mechanics, Information Theory, Probability Theory, Ergodic Theory and the Theory of Dynamical Systems. Specifically, we are referring to the classical entropies: the Boltzmann-Gibbs, von Neumann, Shannon, Kolmogorov-Sinai and topological entropies. In addition to their common name, which is historically justified (as we briefly describe in this review), other commonality of the classical entropies is the important role that they have played and are still playing in the theory and applications of their respective fields and beyond. Therefore, it is not surprising that, in the course of time, many other instances of the overarching concept of entropy have been proposed, most of them tailored to specific purposes. Following the current usage, we will refer to all of them, whether classical or new, simply as entropies. Precisely, the subject of this review is their applications in data analysis and machine learning. The reason for these particular applications is that entropies are very well suited to characterize probability mass distributions, typically generated by finite-state processes or symbolized signals. Therefore, we will focus on entropies defined as positive functionals on probability mass distributions and provide an axiomatic characterization that goes back to Shannon and Khinchin. Given the plethora of entropies in the literature, we have selected a representative group, including the classical ones. The applications summarized in this review finely illustrate the power and versatility of entropy in data analysis and machine learning.

Discovering physical laws from data is a fundamental challenge in scientific research, particularly when high-quality data are scarce or costly to obtain. Traditional methods for identifying dynamical systems often struggle with noise sensitivity, inefficiency in data usage, and the inability to quantify uncertainty effectively. To address these challenges, we propose Langevin-Assisted Active Physical Discovery (LAPD), a Bayesian framework that integrates replica-exchange stochastic gradient Langevin Monte Carlo to simultaneously enable efficient system identification and robust uncertainty quantification (UQ). By balancing gradient-driven exploration in coefficient space and generating an ensemble of candidate models during exploitation, LAPD achieves reliable, uncertainty-aware identification with noisy data. In the face of data scarcity, the probabilistic foundation of LAPD further promotes the integration of active learning (AL) via a hybrid uncertainty-space-filling acquisition function. This strategy sequentially selects informative data to reduce data collection costs while maintaining accuracy. We evaluate LAPD on diverse nonlinear systems such as the Lotka-Volterra, Lorenz, Burgers, and Convection-Diffusion equations, demonstrating its robustness with noisy and limited data as well as superior uncertainty calibration compared to existing methods. The AL extension reduces the required measurements by around 60% for the Lotka-Volterra system and by around 40% for Burgers' equation compared to random data sampling, highlighting its potential for resource-constrained experiments. Our framework establishes a scalable, uncertainty-aware methodology for data-efficient discovery of dynamical systems, with broad applicability to problems where high-fidelity data acquisition is prohibitively expensive.