Distances between probability distributions are a key component of many statistical machine learning tasks, from two-sample testing to generative modeling, among others. We introduce a novel distance between measures that compares them through a Schatten norm of their kernel covariance operators. We show that this new distance is an integral probability metric that can be framed between a Maximum Mean Discrepancy (MMD) and a Wasserstein distance. In particular, we show that it avoids some pitfalls of MMD, by being more discriminative and robust to the choice of hyperparameters. Moreover, it benefits from some compelling properties of kernel methods, that can avoid the curse of dimensionality for their sample complexity. We provide an algorithm to compute the distance in practice by introducing an extension of kernel matrix for difference of distributions that could be of independent interest. Those advantages are illustrated by robust approximate Bayesian computation under contamination as well as particle flow simulations.

Unlike classification, whose goal is to estimate the class of each data point in a dataset, prevalence estimation or quantification is a task that aims to estimate the distribution of classes in a dataset. The two main tasks in prevalence estimation are to adjust for bias, due to the prevalence in the training dataset, and to quantify the uncertainty in the estimate. The standard methods used to quantify uncertainty in prevalence estimates are bootstrapping and Bayesian quantification methods. It is not clear which approach is ideal in terms of precision (i.e. the width of confidence intervals) and coverage (i.e. the confidence intervals being well-calibrated). Here, we propose Precise Quantifier (PQ), a Bayesian quantifier that is more precise than existing quantifiers and with well-calibrated coverage. We discuss the theory behind PQ and present experiments based on simulated and real-world datasets. Through these experiments, we establish the factors which influence quantification precision: the discriminatory power of the underlying classifier; the size of the labeled dataset used to train the quantifier; and the size of the unlabeled dataset for which prevalence is estimated. Our analysis provides deep insights into uncertainty quantification for quantification learning.

In scientific domains -- from biology to the social sciences -- many questions boil down to \textit{What effect will we observe if we intervene on a particular variable?} If the causal relationships (e.g.~a causal graph) are known, it is possible to estimate the intervention distributions. In the absence of this domain knowledge, the causal structure must be discovered from the available observational data. However, observational data are often compatible with multiple causal graphs, making methods that commit to a single structure prone to overconfidence. A principled way to manage this structural uncertainty is via Bayesian inference, which averages over a posterior distribution on possible causal structures and functional mechanisms. Unfortunately, the number of causal structures grows super-exponentially with the number of nodes in the graph, making computations intractable. We propose to circumvent these challenges by using meta-learning to create an end-to-end model: the Model-Averaged Causal Estimation Transformer Neural Process (MACE-TNP). The model is trained to predict the Bayesian model-averaged interventional posterior distribution, and its end-to-end nature bypasses the need for expensive calculations. Empirically, we demonstrate that MACE-TNP outperforms strong Bayesian baselines. Our work establishes meta-learning as a flexible and scalable paradigm for approximating complex Bayesian causal inference, that can be scaled to increasingly challenging settings in the future.

We propose Bayesian Hierarchical Invariant Prediction (BHIP) reframing Invariant Causal Prediction (ICP) through the lens of Hierarchical Bayes. We leverage the hierarchical structure to explicitly test invariance of causal mechanisms under heterogeneous data, resulting in improved computational scalability for a larger number of predictors compared to ICP. Moreover, given its Bayesian nature BHIP enables the use of prior information. In this paper, we test two sparsity inducing priors: horseshoe and spike-and-slab, both of which allow us a more reliable identification of causal features. We test BHIP in synthetic and real-world data showing its potential as an alternative inference method to ICP.

Transfer learning has become an essential paradigm in artificial intelligence, enabling the transfer of knowledge from a source task to improve performance on a target task. This approach, particularly through techniques such as pretraining and fine-tuning, has seen significant success in fields like computer vision and natural language processing. However, despite its widespread use, how to reliably assess the transferability of knowledge remains a challenge. Understanding the theoretical underpinnings of each transferability metric is critical for ensuring the success of transfer learning. In this survey, we provide a unified taxonomy of transferability metrics, categorizing them based on transferable knowledge types and measurement granularity. This work examines the various metrics developed to evaluate the potential of source knowledge for transfer learning and their applicability across different learning paradigms emphasizing the need for careful selection of these metrics. By offering insights into how different metrics work under varying conditions, this survey aims to guide researchers and practitioners in selecting the most appropriate metric for specific applications, contributing to more efficient, reliable, and trustworthy AI systems. Finally, we discuss some open challenges in this field and propose future research directions to further advance the application of transferability metrics in trustworthy transfer learning.

This monograph, spanning three chapters, explores Inverse Reinforcement Learning (IRL). The first two chapters view inverse reinforcement learning (IRL) through the lens of revealed preferences from microeconomics while the third chapter studies adaptive IRL via Langevin dynamics stochastic gradient algorithms. Chapter uses classical revealed preference theory (Afriat's theorem and extensions) to identify constrained utility maximizers based on observed agent actions. This allows for the reconstruction of set-valued estimates of an agent's utility. We illustrate this procedure by identifying the presence of a cognitive radar and reconstructing its utility function. The chapter also addresses the construction of a statistical detector for utility maximization behavior when agent actions are corrupted by noise. Chapter 2 studies Bayesian IRL. It investigates how an analyst can determine if an observed agent is a rationally inattentive Bayesian utility maximizer (i.e., simultaneously optimizing its utility and observation likelihood). The chapter discusses inverse stopping-time problems, focusing on reconstructing the continuation and stopping costs of a Bayesian agent operating over a random horizon. We then apply this IRL methodology to identify the presence of a Bayes-optimal sequential detector. Additionally, Chapter 2 provides a concise overview of discrete choice models, inverse Bayesian filtering, and inverse stochastic gradient algorithms for adaptive IRL. Finally, Chapter 3 introduces an adaptive IRL approach utilizing passive Langevin dynamics. This method aims to track time-varying utility functions given noisy and misspecified gradients. In essence, the adaptive IRL algorithms presented in Chapter 3 can be conceptualized as inverse stochastic gradient algorithms, as they learn the utility function in real-time while a stochastic gradient algorithm is in operation.

We study a sequential decision-making problem motivated by recent regulatory and technological shifts that limit access to individual user data in recommender systems (RSs), leaving only population-level preference information. This privacy-aware setting poses fundamental challenges in planning under uncertainty: Effective personalization requires exploration to infer user preferences, yet unsatisfactory recommendations risk immediate user churn. To address this, we introduce the Rec-APC model, in which an anonymous user is drawn from a known prior over latent user types (e.g., personas or clusters), and the decision-maker sequentially selects items to recommend. Feedback is binary -- positive responses refine the posterior via Bayesian updates, while negative responses result in the termination of the session. We prove that optimal policies converge to pure exploitation in finite time and propose a branch-and-bound algorithm to efficiently compute them. Experiments on synthetic and MovieLens data confirm rapid convergence and demonstrate that our method outperforms the POMDP solver SARSOP, particularly when the number of user types is large or comparable to the number of content categories. Our results highlight the applicability of this approach and inspire new ways to improve decision-making under the constraints imposed by aggregated preference data.

Multiobject tracking (MOT) is an important task in applications including autonomous driving, ocean sciences, and aerospace surveillance. Traditional MOT methods are model-based and combine sequential Bayesian estimation with data association and an object birth model. More recent methods are fully data-driven and rely on the training of neural networks. Both approaches offer distinct advantages in specific settings. In particular, model-based methods are generally applicable across a wide range of scenarios, whereas data-driven MOT achieves superior performance in scenarios where abundant labeled data for training is available. A natural thought is whether a general framework can integrate the two approaches. This paper introduces a hybrid method that utilizes neural networks to enhance specific aspects of the statistical model in Bayesian MOT that have been identified as overly simplistic. By doing so, the performance of the prediction and update steps of Bayesian MOT is improved. To ensure tractable computation, our framework uses belief propagation to avoid high-dimensional operations combined with sequential Monte Carlo methods to perform low-dimensional operations efficiently. The resulting method combines the flexibility and robustness of model-based approaches with the capability to learn complex information from data of neural networks. We evaluate the performance of the proposed method based on the nuScenes autonomous driving dataset and demonstrate that it has state-of-the-art performance

W e present the Object-Based Sub-Environment Recognition (OBSER) framework, a novel Bayesian framework that infers three fundamental relationships between sub-environments and their constituent objects. In the OBSER framework, metric and self-supervised learning models estimate the object distributions of sub-environments on the latent space to compute these measures. Both theoretically and empirically, we validate the proposed framework by introducing the ( ϵ, δ) statistically separable (EDS) function which indicates the alignment of the representation. Our framework reliably performs inference in open-world and photorealistic environments and outperforms scene-based methods in chained retrieval tasks. The OBSER framework enables zero-shot recognition of environments to achieve autonomous environment understanding.

Bayesian optimisation in the latent space of a Variational AutoEncoder (VAE) is a powerful framework for optimisation tasks over complex structured domains, such as the space of scientifically interesting molecules. However, existing approaches tightly couple the surrogate and generative models, which can lead to suboptimal performance when the latent space is not tailored to specific tasks, which in turn has led to the proposal of increasingly sophisticated algorithms. In this work, we explore a new direction, instead proposing a decoupled approach that trains a generative model and a Gaussian Process (GP) surrogate separately, then combines them via a simple yet principled Bayesian update rule. This separation allows each component to focus on its strengths -- structure generation from the VAE and predictive modelling by the GP. We show that our decoupled approach improves our ability to identify high-potential candidates in molecular optimisation problems under constrained evaluation budgets.