We consider the problem of two-sample testing in a semi-supervised setting with abundant unlabeled covariate data. Standard two-sample tests neglect covariate information, which has the potential to significantly boost performance. However, incorporating covariates potentially breaks the exchangeability assumption under the null, which further complicates a calibration procedure. To address these issues, we propose a semi-supervised method that produces a test statistic with asymptotic normality, while effectively integrating additional information from covariates. Our test is straightforward to calibrate due to the asymptotic normality under the null and achieves asymptotic power that is often much higher than existing kernel tests without covariates. Furthermore, we formally show that the proposed method is consistent in power against fixed and local alternatives. Simulations confirm the practical and theoretical strengths of our approach.

We study clustered multitask learning in a semiparametric setting where tasks share a latent cluster structure in their target parameters but exhibit heterogeneous, potentially infinite-dimensional nuisance components. Such heterogeneity poses a major challenge for existing multitask learning methods, which typically rely on aligned feature spaces or homogeneous task structures. To address this challenge, we propose an adaptive fused orthogonal estimator that integrates Neyman-orthogonal losses with data-driven pairwise fusion penalties. Our framework leverages task-specific pilot estimates to calibrate the fusion penalties and combines adaptive aggregation with orthogonalization to mitigate the impact of nuisance-parameter estimation error. Theoretically, we show that the proposed estimator achieves exact recovery of the latent clustering with high probability and attains pooled parametric convergence rates proportional to cluster size. Moreover, we establish asymptotic normality and show that, asymptotically, our estimator matches the performance of an oracle procedure that knows the true clustering in advance. Empirically, we show that the proposed method consistently outperforms strong baselines in various simulation setups. A real-world application to U.S. residential energy consumption demonstrates the effectiveness of our approach in uncovering meaningful regional clustering in electricity price elasticity, showcasing the efficacy of our method.

We introduce Mira, a sample-based score for assessing the accuracy of a candidate conditional distribution using only joint samples from the true data-generating process. Relying on the principle that distributions coincide if they assign equal probability mass to all regions, we derive an analytic expression for the Mira statistic, whose average defines the Mira score. This formulation further allows us to compute theoretical reference values and uncertainty estimates when the candidate distribution matches the true one. This framework enables model comparison by quantifying the alignment between the conditional distribution of a candidate model and the true data generating process. Consequently, Mira enables Bayesian model comparison through direct posterior validation, bypassing the challenging evidence computation. We demonstrate its effectiveness across several toy problems and Bayesian inference tasks.

In 2011, Judea Pearl received the Turing Award, considered the Nobel Prize in Computing, for fundamental contributions to artificial intelligence through the development of a calculus for probabilistic and causal reasoning. It includes pioneering the development of causal discovery algorithms. These computer algorithms can analyze large multivariate datasets and automatically discover the causal relationships among the constituent variables. They have been widely used in many critical fields such as medicine and economics to support decisions. However, to our knowledge, they have not been leveraged in law. This paper attempts to alleviate this gap by investigating whether causal discovery algorithms can be leveraged for automated generation of legal arguments. To that end, a novel legal dataset is prepared by identifying 17 legal concepts, such as physical assault and property dispute. A curated collection of 150 homicide cases are annotated with these concepts, e.g., a case is annotated with physical assault only if a physical assault had been reported in that case. Subsequently, a selected set of widely-used causal discovery algorithms is applied to the annotated dataset to discover the causal relationships between the legal concepts. Additionally, the degrees of belief associated with the discovered relationships are quantified in mathematical probabilities. It is shown that some of the causal relationships help generate viable legal arguments, e.g., if one could establish that a physical assault has not taken place during a homicide, it should be a sufficient condition (with probability 1) to establish that the homicide has not been committed due to a property-related dispute. Thus, this paper shows that causal discovery algorithms can be helpful in generating legal arguments, opening up avenues for promising future endeavors.

The Distributional Alignment Game framework provides a powerful variational perspective on Answer-Level Fine-Tuning (ALFT). However, standard algorithms for these games rely on estimating logarithmic rewards from small batches, introducing a systematic bias due to Jensen's inequality that can destabilize training. In this paper, we systematically resolve this structural estimation bias. First, we generalize the alignment game to arbitrary Bregman divergences, showing that for a family of geometries inducing polynomial rewards, we can construct provably exact and unbiased estimators using U-statistics. Second, for the canonical KL divergence game where an exact solution is impossible, we derive a globally robust minimax polynomial estimator that is provably optimal, achieving the fundamental statistical error limit of $Θ(1/K^2)$, which we establish via the Ditzian-Totik theorem. Finally, we synthesize these two approaches to propose a novel Variance-Optimal Augmented Polynomial Optimization Program (AQP) Estimator, proving that by systematically reducing variance, our method achieves not only optimal bias but also provably accelerated game convergence, leading to more efficient and stable training with zero online computational overhead.

Middle-mile logistics describes the problem of routing parcels through a network of hubs, which are linked by a fixed set of trucks. The main challenge comes from the finite capacity of the trucks. The decision to allocate a parcel to a specific truck might block another parcel from using the same truck. It is thus necessary to solve for all parcel routes simultaneously. Exact solution methods scale poorly with the problem size and real-world instances are intractable.

We study the problem of black-box optimization of a function f of any dimension, given function evaluations perturbed by noise. The function is assumed to be locally smooth around one of its global optima, but this smoothness is unknown. Our contribution is an adaptive optimization algorithm, POO or parallel optimistic optimization, that is able to deal with this setting. POO performs almost as well as the best known algorithms requiring the knowledge of the smoothness. Furthermore, POO works for a larger class of functions than what was previously considered, especially for functions that are difficult to optimize, in a very precise sense. We provide a finite-time analysis of POO's performance, which shows that its error after n evaluations is at most a factor of sqrt(ln n) away from the error of the best known optimization algorithms using the knowledge of the smoothness.

Despite being confined within the interior darkness of the skull, the human brain possesses a remarkable ability to interpret, understand and analyse the world out there, plan for unseen futures, and make decisions that can alter the course of events. This extraordinary capability is conjectured to come from the brain's function as a predictive machine, constantly inferring the hidden causes of its sensory inputs to maintain a coherent model of its environment. This view, which dates back to Helmholtz's idea of "perception as unconscious inference" (von Helmholtz, 1866)--evolving into the "Bayesian brain" hypothesis (Doya et al., 2007)--suggests that the brain operates as a constructive statistical organ. It updates its beliefs about the external world based on incoming sensory data under a generative model (GM). The GM furnishes the brain with a structured representation that supports probabilistic beliefs over both the latent dynamical states of the external world, corresponding to the generative process (GP), as well as the observation mappings through which these states give rise to sensory signals. Essentially, the brain continually refines its probabilistic beliefs about both the latent states and the causal mechanisms of the world through a process of online triple estimation, jointly optimising beliefs over: hidden states, model parameters, and their associated uncertainties in accordance with the principles of Bayesian inference (Eells, 2004; Parr et al., 2022). More technically, given a sensory observation yt at time t, perception can be formulated as an online triple estimation scheme, whose three components are: 1) online hidden state inference, 2) online parameter learning, and 3) online uncertainty estimation, all three of which are the core components of our proposed online generalised PC scheme and are elaborated in Section.

Across many scientific disciplines, multiple observations are collected from the same experimental units, and in modern datasets these observations often arise as non-Euclidean random objects. In such settings, the incorporation of random effects is a critical modeling step for efficient estimation and personalized prediction. Although mixed-effects models are well established for scalar outcomes and, more recently, for functional data in Hilbert spaces, general random-effects frameworks for objects in metric spaces remain underdeveloped. In this paper, we propose a nonlinear Fréchet-based algorithm for random-effects modeling of arbitrary random objects defined on a metric space. Using M-estimation theory, we establish conditions under which the proposed metric-space prediction target is consistently estimated under a working random-effects formulation. We then evaluate the empirical performance of the proposed method using both synthetic data and digital health datasets that require practical tools for analyzing random objects in metric spaces, such as multivariate probability distributions and random graphs. We show that, although our method is developed beyond Hilbert spaces, it can outperform existing Hilbert space-based methods.

This chapter introduces the Bayesian reflex -- an analogy with the autonomic nervous system -- as a unifying framework for online learning in AI. Bayesian online algorithms automatically maintain equilibrium in dynamic environments via three mechanisms: belief maintenance through probabilistic representations, sequential updating via Bayes' theorem, and uncertainty-driven action balancing exploration and exploitation. We survey online Bayesian methods, highlighting two computational principles: the look-up table principle for sequential inference in function space, and the ellipsoidal decomposition framework for nearly exact i.i.d. sampling from arbitrary posteriors. These principles are generalized across dynamic emulation, nonparametric state-space models, circular time series, inverse regression for climate model evaluation, and deep architectures via Recursive Gaussian Processes. Decision-making is explored via Thompson sampling and restless bandits. We extend the framework to assess infinite series convergence (applied to climate dynamics and the Riemann Hypothesis), model prime number distributions leading to the discovery of 184 strong Mersenne prime candidates, detect stationarity, and characterize point processes. The Bayesian reflex provides a foundational infrastructure for adaptive AI that continuously learns in a complex world.