Predictive inference is a fundamental task in statistics, traditionally addressed using parametric assumptions about the data distribution and detailed analyses of how models learn from data. In recent years, conformal prediction has emerged as a rapidly growing alternative framework that is particularly well suited to modern applications involving high-dimensional data and complex machine learning models. Its appeal stems from being both distribution-free -- relying mainly on symmetry assumptions such as exchangeability -- and model-agnostic, treating the learning algorithm as a black box. Even under such limited assumptions, conformal prediction provides exact finite-sample guarantees, though these are typically of a marginal nature that requires careful interpretation. This paper explains the core ideas of conformal prediction and reviews selected methods. Rather than offering an exhaustive survey, it aims to provide a clear conceptual entry point and a pedagogical overview of the field.

The company's sudden decision to pull the plug on Sora is a sign of deeper trouble. When I opened Sora this morning, I was met with a flood of strange and disturbing AI-generated videos. On OpenAI's video app, I scrolled through fabricated scenes of the Iran war and a barrage of fake Donald Trumps blabbering about Jeffrey Epstein. In my least favorite clip, I watched a man deep-fry an infant. The app lets users create fairly realistic-looking AI-generated clips--including of their own likeness--and then post them on a TikTok-like feed.

Knowledge distillation has emerged as a powerful technique for compressing large language models (LLMs) into efficient, deployable architectures while preserving their advanced capabilities. Recent advances in low-rank knowledge distillation, particularly methods like Low-Rank Clone (LRC), have demonstrated remarkable empirical success, achieving comparable performance to full-parameter distillation with significantly reduced training data and computational overhead. However, the theoretical foundations underlying these methods remain poorly understood. In this paper, we establish a rigorous theoretical framework for low-rank knowledge distillation in language models. We prove that under mild assumptions, low-rank projection preserves the optimization dynamics, yielding explicit convergence rates of $O(1/\sqrt{T})$. We derive generalization bounds that characterize the fundamental trade-off between model compression and generalization capability, showing that the generalization error scales with the rank parameter as $O(r(m+n)/\sqrt{n})$. Furthermore, we provide an information-theoretic analysis of the activation cloning mechanism, revealing its role in maximizing the mutual information between the teacher's and student's intermediate representations. Our theoretical results offer principled guidelines for rank selection, mathematically suggesting an optimal rank $r^* = O(\sqrt{n})$ where $n$ is the sample size. Experimental validation on standard language modeling benchmarks confirms our theoretical predictions, demonstrating that the empirical convergence, rank scaling, and generalization behaviors align closely with our bounds.

We introduce Exponential Family Discriminant Analysis (EFDA), a unified generative framework that extends classical Linear Discriminant Analysis (LDA) beyond the Gaussian setting to any member of the exponential family. Under the assumption that each class-conditional density belongs to a common exponential family, EFDA derives closed-form maximum-likelihood estimators for all natural parameters and yields a decision rule that is linear in the sufficient statistic, recovering LDA as a special case and capturing nonlinear decision boundaries in the original feature space. We prove that EFDA is asymptotically calibrated and statistically efficient under correct specification, and we generalise it to $K \geq 2$ classes and multivariate data. Through extensive simulation across five exponential-family distributions (Weibull, Gamma, Exponential, Poisson, Negative Binomial), EFDA matches the classification accuracy of LDA, QDA, and logistic regression while reducing Expected Calibration Error (ECE) by $2$-$6\times$, a gap that is structural: it persists for all $n$ and across all class-imbalance levels, because misspecified models remain asymptotically miscalibrated. We further prove and empirically confirm that EFDA's log-odds estimator approaches the Cramér-Rao bound under correct specification, and is the only estimator in our comparison whose mean squared error converges to zero. Complete derivations are provided for nine distributions. Finally, we formally verify all four theoretical propositions in Lean 4, using Aristotle (Harmonic) and OpenGauss (Math, Inc.) as proof generators, with all outputs independently machine-checked by AXLE (Axiom).

We derive posterior contraction rates (PCRs) and finite-sample Bernstein von Mises (BvM) results for non-parametric Bayesian models by extending the diffusion-based framework of Mou et al. (2024) to the infinite-dimensional setting. The posterior is represented as the invariant measure of a Langevin stochastic partial differential equation (SPDE) on a separable Hilbert space, which allows us to control posterior moments and obtain non-asymptotic concentration rates in Hilbert norms under various likelihood curvature and regularity conditions. We also establish a quantitative Laplace approximation for the posterior. The theory is illustrated in a nonparametric linear Gaussian inverse problem.

Generating samples from a continuous probability density is a central algorithmic problem across statistics, engineering, and the sciences. For high-dimensional settings, Hamiltonian Monte Carlo (HMC) is the default algorithm across mainstream software packages. However, despite the extensive line of work on HMC and its widespread empirical success, it remains unclear how many iterations of HMC are required as a function of the dimension $d$. On one hand, a variety of results show that Metropolized HMC converges in $O(d^{1/4})$ iterations from a warm start close to stationarity. On the other hand, Metropolized HMC is significantly slower without a warm start, e.g., requiring $Ω(d^{1/2})$ iterations even for simple target distributions such as isotropic Gaussians. Finding a warm start is therefore the computational bottleneck for HMC. We resolve this issue for the well-studied setting of sampling from a probability distribution satisfying strong log-concavity (or isoperimetry) and third-order derivative bounds. We prove that \emph{non-Metropolized} HMC generates a warm start in $\tilde{O}(d^{1/4})$ iterations, after which we can exploit the warm start using Metropolized HMC. Our final complexity of $\tilde{O}(d^{1/4})$ is the fastest algorithm for high-accuracy sampling under these assumptions, improving over the prior best of $\tilde{O}(d^{1/2})$. This closes the long line of work on the dimensional complexity of MHMC for such settings, and also provides a simple warm-start prescription for practical implementations.

Energy-based models for discrete domains, such as graphs, explicitly capture relative likelihoods, naturally enabling composable probabilistic inference tasks like conditional generation or enforcing constraints at test-time. However, discrete energy-based models typically struggle with efficient and high-quality sampling, as off-support regions often contain spurious local minima, trapping samplers and causing training instabilities. This has historically resulted in a fidelity gap relative to discrete diffusion models. We introduce Graph Energy Matching (GEM), a generative framework for graphs that closes this fidelity gap. Motivated by the transport map optimization perspective of the Jordan-Kinderlehrer-Otto (JKO) scheme, GEM learns a permutation-invariant potential energy that simultaneously provides transport-aligned guidance from noise toward data and refines samples within regions of high data likelihood. Further, we introduce a sampling protocol that leverages an energy-based switch to seamlessly bridge: (i) rapid, gradient-guided transport toward high-probability regions to (ii) a mixing regime for exploration of the learned graph distribution. On molecular graph benchmarks, GEM matches or exceeds strong discrete diffusion baselines. Beyond sample quality, explicit modeling of relative likelihood enables targeted exploration at inference time, facilitating compositional generation, property-constrained sampling, and geodesic interpolation between graphs.

Its low-rank QTT structure arises from intrinsic spectral smoothness in continuous models, or from spectral energy concentration as the number of components D grows in discrete models. We demonstrate this on weighted sums of Bernoulli and lognormal random variables. In the latter, the approach reaches high-resolution discretizations of N = 230 frequency modes on standard hardware, far beyond the N =224 ceiling of dense implementations. These compressed representations enable efficient computation of Value at Risk (VaR) and Expected Shortfall (ES), supporting applications in quantitative finance and beyond. I. INTRODUCTION Weighted sums of independent random variables constitute a basic probabilistic model, describing macroscopic behavior arising from the aggregation of microscopic stochastic components. These models arise in a wide range of applications. Their probability distribution generally lacks a closed-form expression, and their evaluation involves multidimensional convolution integrals that are susceptible to the curse of dimensionality. Consequently, evaluating these models relies on specializednumericalmethods. Whilethese methods have been adapted for discrete settings [18, 19], they are frequently hampered by persistent Gibbs oscillations, which arise from distributional discontinuities and preclude uniform convergence [20, 21]. No existing method simultaneously achieves an accurate approximation of the exact, fully non-Gaussian target distribution while remaining scalable to larger, practically relevant system sizes. In this work, we introduce a new algorithm that combines the Fourier spectral method with tensor-network techniques.

Active learning reduces labeling costs by selecting samples that maximize information gain. A dominant framework, Query-by-Committee (QBC), typically relies on perturbation-based diversity by inducing model disagreement through random feature subsetting or data blinding. While this approximates one notion of epistemic uncertainty, it sacrifices direct characterization of the plausible hypothesis space. We propose the complementary approach: Rashomon Ensembled Active Learning (REAL) which constructs a committee by exhaustively enumerating the Rashomon Set of all near-optimal models. To address functional redundancy within this set, we adopt a PAC-Bayesian framework using a Gibbs posterior to weight committee members by their empirical risk. Leveraging recent algorithmic advances, we exactly enumerate this set for the class of sparse decision trees. Across synthetic and established active learning baselines, REAL outperforms randomized ensembles, particularly in moderately noisy environments where it strategically leverages expanded model multiplicity to achieve faster convergence.

We propose stepwise variational inference (VI) with vine copulas: a universal VI procedure that combines vine copulas with a novel stepwise estimation procedure of the variational parameters. Vine copulas consist of a nested sequence of trees built from copulas, where more complex latent dependence can be modeled with increasing number of trees. We propose to estimate the vine copula approximate posterior in a stepwise fashion, tree by tree along the vine structure. Further, we show that the usual backward Kullback-Leibler divergence cannot recover the correct parameters in the vine copula model, thus the evidence lower bound is defined based on the Rényi divergence. Finally, an intuitive stopping criterion for adding further trees to the vine eliminates the need to pre-define a complexity parameter of the variational distribution, as required for most other approaches. Thus, our method interpolates between mean-field VI (MFVI) and full latent dependence. In many applications, in particular sparse Gaussian processes, our method is parsimonious with parameters, while outperforming MFVI.