Conformal prediction provides distribution-free prediction intervals with finite-sample coverage guarantees, and recent work by Snell \& Griffiths reframes it as Bayesian Quadrature (BQ-CP), yielding powerful data-conditional guarantees via Dirichlet posteriors over thresholds. However, BQ-CP fundamentally requires the i.i.d. assumption -- a limitation the authors themselves identify. Meanwhile, weighted conformal prediction handles distribution shift via importance weights but remains frequentist, producing only point-estimate thresholds. We propose \textbf{Weighted Bayesian Conformal Prediction (WBCP)}, which generalizes BQ-CP to arbitrary importance-weighted settings by replacing the uniform Dirichlet $\Dir(1,\ldots,1)$ with a weighted Dirichlet $\Dir(\neff \cdot \tilde{w}_1, \ldots, \neff \cdot \tilde{w}_n)$, where $\neff$ is Kish's effective sample size. We prove four theoretical results: (1)~$\neff$ is the unique concentration parameter matching frequentist and Bayesian variances; (2)~posterior standard deviation decays as $O(1/\sqrt{\neff})$; (3)~BQ-CP's stochastic dominance guarantee extends to per-weight-profile data-conditional guarantees; (4)~the HPD threshold provides $O(1/\sqrt{\neff})$ improvement in conditional coverage. We instantiate WBCP for spatial prediction as \emph{Geographical BQ-CP}, where kernel-based spatial weights yield per-location posteriors with interpretable diagnostics. Experiments on synthetic and real-world spatial datasets demonstrate that WBCP maintains coverage guarantees while providing substantially richer uncertainty information.

Gaussian process ($GP$) regression is a widely used non-parametric modeling tool, but its cubic complexity in the training size limits its use on massive data sets. A practical remedy is to predict using only the nearest neighbours of each test point, as in Nearest Neighbour Gaussian Process ($NNGP$) regression for geospatial problems and the related scalable $GPnn$ method for more general machine-learning applications. Despite their strong empirical performance, the large-$n$ theory of $NNGP/GPnn$ remains incomplete. We develop a theoretical framework for $NNGP$ and $GPnn$ regression. Under mild regularity assumptions, we derive almost sure pointwise limits for three key predictive criteria: mean squared error ($MSE$), calibration coefficient ($CAL$), and negative log-likelihood ($NLL$). We then study the $L_2$-risk, prove universal consistency, and show that the risk attains Stone's minimax rate $n^{-2α/(2p+d)}$, where $α$ and $p$ capture regularity of the regression problem. We also prove uniform convergence of $MSE$ over compact hyper-parameter sets and show that its derivatives with respect to lengthscale, kernel scale, and noise variance vanish asymptotically, with explicit rates. This explains the observed robustness of $GPnn$ to hyper-parameter tuning. These results provide a rigorous statistical foundation for $NNGP/GPnn$ as a highly scalable and principled alternative to full $GP$ models.

This study presents a conditional flow matching framework for solving physics-constrained Bayesian inverse problems. In this setting, samples from the joint distribution of inferred variables and measurements are assumed available, while explicit evaluation of the prior and likelihood densities is not required. We derive a simple and self-contained formulation of both the unconditional and conditional flow matching algorithms, tailored specifically to inverse problems. In the conditional setting, a neural network is trained to learn the velocity field of a probability flow ordinary differential equation that transports samples from a chosen source distribution directly to the posterior distribution conditioned on observed measurements. This black-box formulation accommodates nonlinear, high-dimensional, and potentially non-differentiable forward models without restrictive assumptions on the noise model. We further analyze the behavior of the learned velocity field in the regime of finite training data. Under mild architectural assumptions, we show that overtraining can induce degenerate behavior in the generated conditional distributions, including variance collapse and a phenomenon termed selective memorization, wherein generated samples concentrate around training data points associated with similar observations. A simplified theoretical analysis explains this behavior, and numerical experiments confirm it in practice. We demonstrate that standard early-stopping criteria based on monitoring test loss effectively mitigate such degeneracy. The proposed method is evaluated on several physics-based inverse problems. We investigate the impact of different choices of source distributions, including Gaussian and data-informed priors. Across these examples, conditional flow matching accurately captures complex, multimodal posterior distributions while maintaining computational efficiency.

In this paper, we derive rates of convergence in the high-dimensional central limit theorem for Polyak-Ruppert averaged iterates generated by the asynchronous Q-learning algorithm with a polynomial stepsize $k^{-ω},\, ω\in (1/2, 1]$. Assuming that the sequence of state-action-next-state triples $(s_k, a_k, s_{k+1})_{k \geq 0}$ forms a uniformly geometrically ergodic Markov chain, we establish a rate of order up to $n^{-1/6} \log^{4} (nS A)$ over the class of hyper-rectangles, where $n$ is the number of samples used by the algorithm and $S$ and $A$ denote the numbers of states and actions, respectively. To obtain this result, we prove a high-dimensional central limit theorem for sums of martingale differences, which may be of independent interest. Finally, we present bounds for high-order moments for the algorithm's last iterate.

Clustering in high-dimensional settings with severe feature noise remains challenging, especially when only a small subset of dimensions is informative and the final number of clusters is not specified in advance. In such regimes, partition recovery, feature relevance learning, and structural adaptation are tightly coupled, and standard likelihood-based methods can become unstable or overly sensitive to noisy dimensions. We propose DIVI, a data-informed variational clustering framework that combines global feature gating with split-based adaptive structure growth. DIVI uses informative prior initialization to stabilize optimization, learns feature relevance in a differentiable manner, and expands model complexity only when local diagnostics indicate underfit. Beyond clustering performance, we also examine runtime scalability and parameter sensitivity in order to clarify the computational and practical behavior of the framework. Empirically, we find that DIVI performs competitively under severe feature noise, remains computationally feasible, and yields interpretable feature-gating behavior, while also exhibiting conservative growth and identifiable failure regimes in challenging settings. Overall, DIVI is best viewed as a practical variational clustering framework for noisy high-dimensional data rather than as a fully Bayesian generative solution.

Deep linear networks (DLNs) are used as an analytically tractable model of the training dynamics of deep neural networks. While gradient descent in DLNs is known to exhibit saddle-to-saddle dynamics, the impact of stochastic gradient descent (SGD) noise on this regime remains poorly understood. We investigate the dynamics of SGD during training of DLNs in the saddle-to-saddle regime. We model the training dynamics as stochastic Langevin dynamics with anisotropic, state-dependent noise. Under the assumption of aligned and balanced weights, we derive an exact decomposition of the dynamics into a system of one-dimensional per-mode stochastic differential equations. This establishes that the maximal diffusion along a mode precedes the corresponding feature being completely learned. We also derive the stationary distribution of SGD for each mode: in the absence of label noise, its marginal distribution along specific features coincides with the stationary distribution of gradient flow, while in the presence of label noise it approximates a Boltzmann distribution. Finally, we confirm experimentally that the theoretical results hold qualitatively even without aligned or balanced weights. These results establish that SGD noise encodes information about the progression of feature learning but does not fundamentally alter the saddle-to-saddle dynamics.

Dr. David Blackwell was a mathematician and statistician of the first rank, whose contributions to statistical theory, game theory, and decision theory predated many of the algorithmic breakthroughs that define modern artificial intelligence. This survey examines three of his most consequential theoretical results the Rao Blackwell theorem, the Blackwell Approachability theorem, and the Blackwell Informativeness theorem (comparison of experiments) and traces their direct influence on contemporary AI and machine learning. We show that these results, developed primarily in the 1940s and 1950s, remain technically live across modern subfields including Markov Chain Monte Carlo inference, autonomous mobile robot navigation (SLAM), generative model training, no-regret online learning, reinforcement learning from human feedback (RLHF), large language model alignment, and information design. NVIDIAs 2024 decision to name their flagship GPU architecture (Blackwell) provides vivid testament to his enduring relevance. We also document an emerging frontier: explicit Rao Blackwellized variance reduction in LLM RLHF pipelines, recently proposed but not yet standard practice. Together, Blackwell theorems form a unified framework addressing information compression, sequential decision making under uncertainty, and the comparison of information sources precisely the problems at the core of modern AI.

Accurate classification requires not only high predictive accuracy but also well-calibrated confidence estimates. Yet, modern deep neural networks (DNNs) are often overconfident, primarily due to overfitting on the negative log-likelihood (NLL). While focal loss variants alleviate this issue, they typically reduce accuracy, revealing a persistent trade-off between calibration and predictive performance. Motivated by the complementary strengths of generative and discriminative classifiers, we propose Generative Cross-Entropy (GCE), which maximizes $p(x|y)$ and is equivalent to cross-entropy augmented with a class-level confidence regularizer. Under mild conditions, GCE is strictly proper. Across CIFAR-10/100, Tiny-ImageNet, and a medical imaging benchmark, GCE improves both accuracy and calibration over cross-entropy, especially in the long-tailed scenario. Combined with adaptive piecewise temperature scaling (ATS), GCE attains calibration competitive with focal-loss variants without sacrificing accuracy.

There is a growing demand for efficient data removal to comply with regulations like the GDPR and to mitigate the influence of biased or corrupted data. This has motivated the field of machine unlearning, which aims to eliminate the influence of specific data subsets without the cost of full retraining. In this work, we propose a statistical framework for machine unlearning with generic loss functions and establish theoretical guarantees. For squared loss, especially, we develop Unlearning Least Squares (ULS) and establish its minimax optimality for estimating the model parameter of remaining data when only the pre-trained estimator, forget samples, and a small subsample of the remaining data are available. Our results reveal that the estimation error decomposes into an oracle term and an unlearning cost determined by the forget proportion and the forget model bias. We further establish asymptotically valid inference procedures without requiring full retraining. Numerical experiments and real-data applications demonstrate that the proposed method achieves performance close to retraining while requiring substantially less data access.

This paper presents a Structured Source-Wise Adaptive Diffusion Framework for linear and nonlinear blind source separation. The framework interprets each latent dimension as a source component and assigns to it an individual adaptive diffusion mechanism, thereby establishing source-wise latent modeling rather than relying on a single shared latent prior. The resulting formulation learns source recovery and the mixing/reconstruction process jointly within a unified end-to-end objective, allowing model parameters and latent sources to adapt simultaneously during training. This yields a common framework for both linear and nonlinear blind source separation. In the present instantiation, each source is further equipped with its own adaptive Gaussian process (GP) prior to impose source-wise temporal structure on the latent trajectories, while the overall framework is not restricted to Gaussian process priors and can in principle accommodate other structured source priors. The proposed model thus provides a general structured diffusion-based route to unsupervised source recovery, with potential relevance beyond blind source separation to interpretable latent modeling, source-wise disentanglement, and potentially identifiable nonlinear latent-variable learning under appropriate structural conditions.