We ask: when do Bayesian model averaging (BMA) weights over decision trees carry sufficient epistemic information to justify committed exploitation of the averaging distribution? We answer this question in closed form for Bayesian decision trees (BDTs) with Dirichlet-Multinomial leaf models and a Catalan-exponential tree-size prior (Schetinin&Jakaite, 2025), establishing a complete non-asymptotic theory of rational commitment thresholds.

Spectral clustering is a tool for extracting meaningful information from data by grouping similar objectsDtogether [1]. The method uses the eigenvector of an adjacency matrix for embedding the data into a space that captures the underlying group structure [2]. High-dimensional signals, magnetic resonance images, and hyperspectral images can be costly to acquire; even simple direct comparisons could be infeasible among such data sets. Our work shows that the meaningful organization extracted from spectral clustering is preserved under the perturbation from making compressed, incomplete and inaccurate measurements. Using bounds on the perturbation of eigenvectors, we establish error bounds of the spectral embedding when matrix completion and compressed sensing measurements are used. Given some error Nǫ in the entries of an affinity matrix A RN N, we show that the space spanned by the first k eigenvector are all within O(Nǫ) of the span of the unperturbed eigenvectors. We prove that the perturbed spectral coordinates are within O(Nǫ)of a unitary transform of the unperturbed coordinates and can give k-means cluster assignments within O(Nǫ) of the unperturbed case. This analysis holds true when the error perturbation in the entries of an affinity matrix |A(i,j) A (i,j)| ǫ is caused from making compressed arXiv:1011.0997v1

In this paper we study the performance of the Projected Gradient Descent(PGD) algorithm for $\ell_{p}$-constrained least squares problems that arise in the framework of Compressed Sensing. Relying on the Restricted Isometry Property, we provide convergence guarantees for this algorithm for the entire range of $0\leq p\leq1$, that include and generalize the existing results for the Iterative Hard Thresholding algorithm and provide a new accuracy guarantee for the Iterative Soft Thresholding algorithm as special cases. Our results suggest that in this group of algorithms, as $p$ increases from zero to one, conditions required to guarantee accuracy become stricter and robustness to noise deteriorates.

In a nutshell, the outcomes ought to be indistinguishable from random draws from the predictive distributions. In this paper, we review, extend, and bridge notions of calibration that have been proposed for classification and regression tasks. Particular emphasis is given to hierarchical relations between the various notions, as they apply to general real-valued data, continuous outcomes, count data, nominal classes, and binary outcomes. To highlight a number of contributions, we introduce the notion of modal calibration for nominal outcomes, we distinguish full, partial, and average calibration in this setting, and we show that double probability integral transform (PIT) calibration is logically independent of previously proposed concepts of calibration for discrete outcomes. Furthermore, we generalize extant results on concepts of calibration that are expressed in terms of properties or functionals of the predictive distributions, such as means, quantiles, or event probabilities. Throughout the paper, we illustrate the concepts and their hierarchical relations in worked examples, and we provide algorithmic tools that support the construction of instructive examples and counterexamples. Keywords: Auto-calibration, confidence calibration, diagnostic evaluation of probabilistic predictions, distributional properties, probability integral transform (PIT), reliability.

This paper investigates non-stationary online learning using the metric of interval regret, which requires an online algorithm to perform well over every time interval. We propose the first online learning algorithm that achieves an interval regret bound scaling with gradient variation, a fundamental measure of the cumulative change in online function gradients, which relates to various problem-dependent quantities and is closely connected to stochastic optimization and other problems. Our method employs a simple and efficient two-layer online ensemble structure that achieves strong theoretical guarantees. Specifically, it enjoys a regret bound that simultaneously adapts to various problem-dependent quantities while also preserving the minimax-optimal rate in the worst case. Moreover, recognizing the challenge of hyperparameter tuning, we introduce a Lipschitz- and smoothness-agnostic variant that automatically adapts to these potentially unknown constants. This is primarily enabled by a novel Lipschitz-adaptive meta algorithm, which may be of independent interest. Beyond interval regret, our method also yields broader implications: it provides versatile bounds for interval dynamic regret, a stronger measure that competes with changing comparators over any interval, and yields the first piecewise characterization for stochastic extended adversarial optimization. Theoretical findings are validated by experiments.

Given a set of mixtures, blind source separation attempts to retrieve the source signals without or with very little information of the the mixing process. We present a geometric approach for blind separation of nonnegative linear mixtures termed {\em facet component analysis} (FCA). The approach is based on facet identification of the underlying cone structure of the data. Earlier works focus on recovering the cone by locating its vertices (vertex component analysis or VCA) based on a mutual sparsity condition which requires each source signal to possess a stand-alone peak in its spectrum. We formulate alternative conditions so that enough data points fall on the facets of a cone instead of accumulating around the vertices. To find a regime of unique solvability, we make use of both geometric and density properties of the data points, and develop an efficient facet identification method by combining data classification and linear regression. For noisy data, we show that denoising methods may be employed, such as the total variation technique in imaging processing, and principle component analysis. We show computational results on nuclear magnetic resonance spectroscopic data to substantiate our method.

Large Language Models remain plagued by hallucinations. Recent work has sought to tame their prevalence using statistical techniques based on conformal prediction, with both theoretical and empirical success. However, these methods operate in a post-hoc fashion, treating the sampling procedure itself as atomic and then surgically altering samples to remove hallucinated claims. This disconnect between filtering and generation can result in samples that are incoherent, inconsistent, or simply unlikely under the model itself. Moreover, post-hoc surgery is unable to shift probability mass towards more useful and helpful responses. To address these issues, we propose to instead sample from approximations to an LLM posterior, where the conditioning event corresponds to a calibrated, high-scoring region. We develop a calibration procedure tailored to the setting of conditional sequential generation that effectively identifies this region and achieves target risk control. Empirically, we apply our method to case studies focused on open-ended biography generation and mathematical problem solving; compared to prior work, we obtain the same statistical guarantees, with higher downstream utility.

This is a month to look out for some powerful new books, with authors taking on challenges of all sorts and imagining whole new worlds. There are fresh ways to think about a cancer diagnosis, a book tackling the real inner world of hormones, in which we are all hormonal all the time, plus a major re-envisioning of the natural world where we abandon the shallows of competition for the depth and intricacies of connection and togetherness. It's quite hard going to get an up-to-date grip on human evolution, even for the best-briefed adult, so a book with sophisticated text and excellent illustrations and diagrams can only be a good thing. Especially if it is curated and edited by Alice Roberts, biological anthropologist, palaeopathologist, broadcaster - and professor of public engagement in science at the University of Birmingham, UK. She worked with a generous-sized international team of experts in many fields of human evolution, including archaeology, palaeontology, anthropology and cognitive science.

Quivering Karmelo Anthony is convicted of murdering Austin Metcalf, 17... but now prosecutors have granted him Hail Mary that could see him jailed for as little as TWO YEARS Trump's $70B immigration crackdown passes the House as sneaky loophole allows $1.8B weaponization'slush fund' to survive I watched footage of the race crime that split America. She's always by Trump's side, trusted with the White House's biggest secrets... and she influences millions Trump ERUPTS behind closed doors as top Republican pleads with him to axe Tulsi Gabbard's spy-chief replacement Leaked transcript of UNAIRED 60 Minutes interview exposes REAL reason'callous' CBS star Scott Pelley'deserved to be fired' Epstein's massage fixer looks PETRIFIED as she's dragged into explosive congressional grilling - and reveals jaw-dropping'blackmail' theory Caitlyn Jenner biographer and Robin Riker's ex William Hasley found dead on hiking trail at 78 Eva Longoria reunites with ex Tony Parker 15 years after cheating scandal split... as shocked fans react Shamed ex mayor Misty Roberts is sentenced to 90 DAYS as she's branded a'predator with hair extensions' by enraged mother of 17-year-old sex assault victim My compulsive bathroom habit that so many are guilty of left me in excruciating pain. DR STUART reveals early signs... cures that work in days... and when to worry Inside Travis Kelce's plan to become'the Shaq of the NFL' after wedding Taylor Swift Moment Real Housewives star Lenny Hochstein's sexual assault accuser'dances' as she leaves Star Island mansion - before filing $100k civil lawsuit Madonna's wild sex claim about JFK Jr now draws surprising response from his outspoken nephew Jack Schlossberg Zodiac killer case takes bombshell turn as unsolved cipher is CRACKED... and America's top codebreakers say evidence is all pointing to one man Kennedy heir Jack Schlossberg concedes Trump is a'genius' as aspiring congressman reveals what he'deeply respects' about president'Great' mom, 32, tried to gas herself and her three young kids to death after inviting them to'popcorn sleepover' in car, prosecutors allege Want to lose up to a stone in six weeks, plus boost your mood and energy levels? Fermented foods from kefir to kombucha are having a moment, hailed for their gut health benefits. But experts say we could be overlooking one of the healthiest ferments out there: sauerkraut.

Stochastic gradient Langevin dynamics combined with Gibbs updates (SGLD--Gibbs) provides a highly scalable approach to approximate Bayesian inference in latent variable models. However, it remains unclear how to tune the algorithm's hyperparameters in a principled manner to ensure the uncertainty estimates are statistically meaningful. In this work, we address this gap in tuning guidance by developing a statistical scaling limit theory for SGLD--Gibbs. We derive a joint asymptotic limit for the global parameters and latent variables under appropriate space-time rescaling. We show that global parameters converge to a diffusion-type limit, while each latent variable converges to a jump process, reflecting the use of intermittent Gibbs updates. This joint jump-diffusion structure reveals how latent-variable randomness contributes to the stationary distribution of the global parameters. We leverage our results to propose explicit guidance on hyperparameter tuning for SGLD--Gibbs that ensures meaningful uncertainty quantification. Numerical experiments show that SGLD--Gibbs with our tuning guidance leads to better parameter estimates, uncertainty quantification, and predictive performance than stochastic variational inference.