Probability & Statistics
Probability theory, stochastic processes, and mathematical statistics
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Counterfactual Spaces
We mathematically axiomatise the stochastics of counterfactuals, by introducing two related frameworks, called counterfactual probability spaces and counterfactual causal spaces, which we collectively term counterfactual spaces. They are, respectively, probability and causal spaces whose underlying measurable spaces are products of world-specific measurable spaces. In contrast to more familiar accounts of counterfactuals founded on causal models, we do not view interventions as a necessary component of a theory of counterfactuals. As an alternative to Pearl's celebrated ladder of causation, we view counterfactuals and interventions are orthogonal concepts, respectively mathematised in counterfactual probability spaces and causal spaces. The two concepts are then combined to form counterfactual causal spaces. At the heart of our theory is the notion of shared information between the worlds, encoded completely within the probability measure and causal kernels, and whose extremes are characterised by independence and synchronisation of worlds. Compared to existing frameworks, counterfactual spaces enable the mathematical treatment of a strictly broader spectrum of counterfactuals.
2601.00507
Jan 2026Statistics Theory
Voronoi Percolation: Topological Stability and Giant Cycles
We study the topological stability of Voronoi percolation in higher dimensions. We show that slightly increasing p allows a discretization that preserves increasing topological properties with high probability. This strengthens a theorem of Bollobás and Riordan and generalizes it to higher dimensions. As a consequence, we prove a sharp phase transition for the emergence of i-dimensional giant cycles in Voronoi percolation on the 2i-dimensional torus.
2601.00793
Jan 2026Probability
2512.24152Score-based sampling without diffusions: Guidance from a simple and modular scheme
Sampling based on score diffusions has led to striking empirical results, and has attracted considerable attention from various research communities. It depends on availability of (approximate) Stein score functions for various levels of additive noise. We describe and analyze a modular scheme that reduces score-based sampling to solving a short sequence of ``nice'' sampling problems, for which high-accuracy samplers are known. We show how to design forward trajectories such that both (a) the terminal distribution, and (b) each of the backward conditional distribution is defined by a strongly log concave (SLC) distribution. This modular reduction allows us to exploit \emph{any} SLC sampling algorithm in order to traverse the backwards path, and we establish novel guarantees with short proofs for both uni-modal and multi-modal densities. The use of high-accuracy routines yields $\varepsilon$-accurate answers, in either KL or Wasserstein distances, with polynomial dependence on $\log(1/\varepsilon)$ and $\sqrt{d}$ dependence on the dimension.
2512.24152
Dec 2025Statistics Theory
2512.23425A general framework for deep learning
This paper develops a general approach for deep learning for a setting that includes nonparametric regression and classification. We perform a framework from data that fulfills a generalized Bernstein-type inequality, including independent, $φ$-mixing, strongly mixing and $\mathcal{C}$-mixing observations. Two estimators are proposed: a non-penalized deep neural network estimator (NPDNN) and a sparse-penalized deep neural network estimator (SPDNN). For each of these estimators, bounds of the expected excess risk on the class of Hölder smooth functions and composition Hölder functions are established. Applications to independent data, as well as to $φ$-mixing, strongly mixing, $\mathcal{C}$-mixing processes are considered. For each of these examples, the upper bounds of the expected excess risk of the proposed NPDNN and SPDNN predictors are derived. It is shown that both the NPDNN and SPDNN estimators are minimax optimal (up to a logarithmic factor) in many classical settings.
2512.23425
Dec 2025Statistics Theory
Basic Inequalities for First-Order Optimization with Applications to Statistical Risk Analysis
We introduce \textit{basic inequalities} for first-order iterative optimization algorithms, forming a simple and versatile framework that connects implicit and explicit regularization. While related inequalities appear in the literature, we isolate and highlight a specific form and develop it as a well-rounded tool for statistical analysis. Let $f$ denote the objective function to be optimized. Given a first-order iterative algorithm initialized at $θ_0$ with current iterate $θ_T$, the basic inequality upper bounds $f(θ_T)-f(z)$ for any reference point $z$ in terms of the accumulated step sizes and the distances between $θ_0$, $θ_T$, and $z$. The bound translates the number of iterations into an effective regularization coefficient in the loss function. We demonstrate this framework through analyses of training dynamics and prediction risk bounds. In addition to revisiting and refining known results on gradient descent, we provide new results for mirror descent with Bregman divergence projection, for generalized linear models trained by gradient descent and exponentiated gradient descent, and for randomized predictors. We illustrate and supplement these theoretical findings with experiments on generalized linear models.
2512.24999
Dec 2025Statistics Theory
Universal entrywise eigenvector fluctuations in delocalized spiked matrix models and asymptotics of rounded spectral algorithms
We consider the distribution of the top eigenvector $\widehat{v}$ of a spiked matrix model of the form $H = θvv^* + W$, in the supercritical regime where $H$ has an outlier eigenvalue of comparable magnitude to $\|W\|$. We show that, if $v$ is sufficiently delocalized, then the distribution of the individual entries of $\widehat{v}$ (not, we emphasize, merely the inner product $\langle \widehat{v}, v\rangle$) is universal over a large class of generalized Wigner matrices $W$ having independent entries, depending only on the first two moments of the distributions of the entries of $W$. This complements the observation of Capitaine and Donati-Martin (2018) that these distributions are not universal when $v$ is instead sufficiently localized. Further, for $W$ having entrywise variances close to constant and thus resembling a Wigner matrix, we show by comparing to the case of $W$ drawn from the Gaussian orthogonal or unitary ensembles that averages of entrywise functions of $\widehat{v}$ behave as they would if $\widehat{v}$ had Gaussian fluctuations around a suitable multiple of $v$. We apply these results to study spectral algorithms followed by rounding procedures in dense stochastic block models and synchronization problems over the cyclic and circle groups, obtaining the first precise asymptotic characterizations of the error rates of such algorithms.
2512.11785
Dec 2025Probability
Can a Higher Order Markov Chain Be Treated as a First Order Markov Chain?
It is well known that any higher order Markov chain can be associated with a first order Markov chain. In this primarily expository article, we present the first fairly comprehensive analysis of the relationship between higher order and first order Markov chains, together with illustrative examples. Our main objective is to address the central question as posed in the title.
2512.01969
Dec 2025Probability
