Comparing conditional distributions is a fundamental challenge in statistics and machine learning, with applications across a wide range of domains. While proposed methods for measuring discrepancies using kernel embeddings of distributions in a reproducing kernel Hilbert space (RKHS) provide powerful non-parametric techniques, the existing literature remains fragmented and lacks a unified theoretical treatment. This paper addresses this gap by establishing a coherent framework for studying kernel-based methods to measure divergence between conditional distributions through what we refer to as conditional maximum mean discrepancy (CMMD). The CMMD consists of a family of metrics which we call levels, with three special cases each using a different type of RKHS embedding: CMMD$_0$ (conditional mean operators), CMMD$_1$ (conditional mean embeddings), and CMMD$_2$ (joint mean embeddings). We additionally introduce a general level $s$ CMMD, clarifying the required assumptions, and establishing mathematical connections between the levels through the lens of operator-based smoothing. In addition to reviewing previously proposed estimators, we introduce a novel doubly robust estimator for the CMMD that maintains consistency provided at least one of the underlying models is correctly specified. We provide numerical experiments demonstrating that the CMMD effectively captures complex conditional dependencies for statistical testing.

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.

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.

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.

Groundwater in the Densu Basin is increasingly threatened by heavy metal contamination, but conventional methods fail to capture the statistical complexity and spatial heterogeneity of pollution indicators. A key challenge is modelling the Heavy Metal Pollution Index (HPI), which is typically skewed and affected by correlated contaminants, leading to biased predictions without transformation. This study develops a predictive framework integrating response transformations with nested cross-validated ensemble machine learning. Three transformations (raw, log, and Gaussian copula) were applied to HPI and evaluated across six learners: support vector regression (SVM), $k$-nearest neighbours (k-NN), CART, Elastic Net, kernel ridge regression, and a stacked Lasso ensemble. Raw-scale models produced deceptively high fits (Elastic Net and stacked ensemble $R^2 \approx 1.0$), suggesting over-optimism. The log transformation stabilised variance (SVM: $R^2 = 0.93$, RMSE $= 0.18$; k-NN: $R^2 = 0.92$, RMSE $= 0.20$). The Gaussian copula gave the most reliable results: stacked ensemble $R^2 = 0.96$ (RMSE $= 0.19$), with other learners maintaining high accuracy. Copula-based models improved residuals and produced spatially plausible maps. DBSCAN clustering revealed Fe and Mn as primary HPI contributors, consistent with regional hydrogeochemistry. Limitations include reliance on random (not spatial) cross-validation and basin-specific scope. Future work should explore spatial validation and other geological settings. Overall, distribution-aware ensembles with clustering diagnostics offer robust, interpretable assessments of groundwater contamination.

Despite rapid recent advances in quantum machine learning, the field is in many ways stuck. Existing approaches can exhibit serious limitations, and we still lack learning frameworks that are simple, interpretable, scalable, and naturally suited to quantum data. To address this, here we introduce quantum Gaussian processes, a Bayesian framework for learning from quantum systems through priors over unknown quantum transformations. We show that, under suitable conditions, unitary quantum stochastic processes define Gaussian processes, thereby enabling regression, classification, and Bayesian optimization directly on quantum data. The key ingredient in this framework is sufficient knowledge of a quantum process's structure and symmetries to define an informative prior through its corresponding quantum kernel, effectively injecting a strong, physics-informed inductive bias into the learning model. We then prove that matchgate, or free-fermionic, evolutions give rise to provable and scalable quantum Gaussian processes, providing the first family in our framework where the unknown unitary acts non-trivially on all qubits. Finally, we demonstrate accurate long-range extrapolation, phase-diagram learning in many-body systems, and sample-efficient Bayesian optimization in a quantum sensing task. Our results identify quantum Gaussian processes as a promising route toward simpler and more structured forms of quantum learning.

Double-machine-learning pipelines for the Average Dose-Response Function rely on kernel-weighted local-linear smoothers, which inherit unbounded functional influence: a single outlier within a kernel window biases the curve across the entire window. We introduce SHIFT (Self-calibrated Heavy-tail Inlier-Fit with Tempering), a robust DML estimator combining cross-fit nuisance orthogonalization with a kernel-local Welsch-loss second stage optimized by Graduated Non-Convexity, and -- the principal design choice -- a defensive OLS refit whose inlier cutoff is scaled by post-GNC residual MAD rather than the raw-outcome MAD. On a localized-contamination stress test at $p=0.25$ this design choice drops level-RMSE from 1.03 to 0.33 while leaving clean and uniformly-contaminated runs unchanged. Across 1,400 main-sweep fits, SHIFT has competitive worst-case shape recovery (RMSE $0.325$ at $p=0.25$, second to Huber-DML's $0.276$); among the three methods with worst-case RMSE below $0.35$, only SHIFT emits a non-uniform per-sample weight vector, recovering the ground-truth outlier mask at mean $F_1 \approx 0.96$ (range $0.945$--$0.968$) on Gaussian-jump DGPs. We pair the estimator with a six-technique Extreme Value Theory diagnostic suite (Hill, GPD-MLE/PWM, GEV, Mean Excess, parameter stability, causal tail coefficient) that lets a practitioner distinguish Frechet from Weibull regimes and choose between SHIFT and L1 alternatives on empirical grounds. Extensions to binary-treatment CATE (Huber pseudo-outcome X-Learner) and time-series ADRF (block-CV + rolling MAD) are included. A counter-intuitive ablation: linear nuisance models (Ridge, Lasso) outperform gradient-boosted nuisances for robust DML under uniform contamination, inverting the usual more-flexible-is-better heuristic.

Many machine learning systems make constrained decisions by optimizing factorized objectives, but the context-specific objective is often treated as fixed. We study contextual decision-weight learning: from logged decisions and proxy outputs, learn an optimizer-facing weight vector w(x) over interpretable decision factors z(x,d), rather than a direct policy or generic predictive score. We propose OTSS, an output-targeted soft-segmentation model that deploys the personalized decision-ready weight vector. At the function-class level, the theory highlights a hard-versus-soft distinction. Hard partitions incur an approximation-estimation tradeoff under overlap, while a realizable fixed-K soft class removes the hard-partition approximation floor and attains a parametric rate. We evaluate OTSS in controlled benchmarks with finite evaluation libraries, where the true weight vector and downstream regret can be computed exactly. In the representative overlap setting, OTSS attains the lowest mean regret among the comparators, including EM mixture regression, the strongest soft-mixture baseline in our comparison; it matches EM on coefficient recovery while running about two orders of magnitude faster. In a matched K=5 benchmark, OTSS remains competitive under hard-routed truth and improves as heterogeneity becomes softer and sample size grows. On a fixed Complete Journey retail anchor with real household covariates and action geometry, OTSS again achieves the lowest mean-regret point estimate.

Online Conformal Prediction (CP) struggles to balance temporal adaptability and structural stability. Feedback-driven methods (e.g., Adaptive Conformal Inference (ACI)) suffer from systemic marginal under-coverage and high interval variance during abrupt shifts, while temporally discounted Bayesian CP suffers from severe structural lag and uncalibrated interval bloat. We propose State-Adaptive Bayesian Conformal Prediction (SA-BCP) to achieve optimal spatio-temporal decoupling. By gating long-term temporal inertia with spatial kernel-density evidence, SA-BCP proactively expands intervals for recognized historical regimes while maintaining tight efficiency during stable states. We rigorously prove this mechanism's optimality, identifying a minimax bias-variance tradeoff governed by an evidence threshold $K$. Extensive benchmarks on volatile financial datasets (2016--2026), including AMD, Gold, and GBP/USD, demonstrate that SA-BCP consistently minimizes the strictly proper Winkler score across diverse confidence levels. Specifically, SA-BCP resolves the systematic under-coverage inherent to ACI variants while simultaneously reducing the uncalibrated interval bloat of Bayesian CP by 10\% to 37\% under high-confidence requests. By elegantly navigating this tradeoff, SA-BCP achieves an optimal balance between conditional reliability and predictive efficiency.

Gradient Boosting Decision Trees (GBDTs) dominate tabular machine learning, with modern implementations like XGBoost, LightGBM, and CatBoost being based on Newton boosting: a second-order descent step in the space of decision trees. Despite its empirical success, the global convergence of Newton boosting is poorly understood compared to first-order boosting. In this paper, we introduce Restricted Newton Descent, which studies convex optimization with Newton's method on Hilbert spaces with inexact iterates, based on the concepts of cosine angle and weak gradient edge. Within this framework, we recover Newton boosting with GBDTs and classical finite-dimensional theory as special cases. We first prove that vanilla Newton boosting achieves a linear rate of convergence for smooth, strongly convex losses that satisfy a Hessian-dominance condition. To handle general convex losses with Lipschitz Hessians, we extend a recent gradient regularized Newton scheme to the restricted weak learner setting. This scheme minimally modifies the classical algorithm by introducing an adaptive $\ell_2$-regularization term proportional to the square root of the gradient norm at each iteration. We establish a $\mathcal{O}(\frac{1}{k^2})$ rate for this scheme, thereby obtaining a globally convergent second-order GBDT algorithm with a rate matching that of first-order boosting with Nesterov momentum. In numerical experiments, we show that our scheme converges while vanilla Newton boosting may diverge.