We consider contraction of Bayesian posterior distributions in nonparametric settings where coefficients of a function over a basis or dictionary are given priors with $p$--exponential tails, including Laplace tails $(p=1)$ and heavier tails $(p<1)$. It is shown that contraction rates improve as $p$ decreases and that full adaptation to smoothness, up to logarithmic factors, is obtained in an appropriate $p\to 0$ regime. As applications, we consider both series priors in white noise regression and shallow ReLU neural networks in random design regression. In particular, we show that overparametrised shallow ReLU networks can adapt to any regularity $0\le β\le 2$. Through a simulation study, we show strong empirical agreement with the behavior predicted by our theory.

We reconcile privacy protection and rate-double-robust inference. The privacy of individuals is protected by a local privacy mechanism: injecting noise into their sensitive data, revealing only the noisy data for inference. Hence, privacy protection hinders inference. In contrast, the inference of a target parameter is rate-double-robust when the large-sample bias of an estimator of the parameter is characterised by a trade-off between the estimation errors of two other, nuisance, parameters. Hence, rate-double-robustness facilitates inference. Our starting point of reconciliation is a class of rate-double-robust target parameters indexed linearly by an infinite-dimensional and nonlinearly by a low-dimensional regression. Among others, this includes causal parameters. To infer these targets privately, we show how suitable privacy mechanisms transfer the semiparametric properties of the sensitive-data model to the private setting. Rate-double-robustness is transferred, enabling locally-private, unbiased and semiparametrically efficient inference of our target parameters. Finally, we transform general nonparametric nuisance estimators into private ones, which inherit convergence properties of their nonprivate counterparts. For parametric nuisance models, we develop a private method-of-moments estimator and its large-sample inference theory.

We develop a partial differential equation formulation of the Behrens-Fisher problem for two independent normal samples with unknown and unequal variances. An orthogonal decomposition separates mean and residual components (corresponding to the centered within-sample variation left after removal of the mean directions) and recasts the studentized difference of sample means as a scale-invariant geometric constraint. This reduction transforms the distributional problem into the evaluation of spherical wedge probabilities, which are identified with harmonic measure and with the value at the origin of a Laplace-Dirichlet boundary value problem. From this framework, we derive exact finite-sample representations for the cumulative distribution function and the probability density function in terms of beta functions, with dependence only on the sample sizes and the variance ratio. These representations place the Behrens-Fisher law in a standard special-function form that is directly accessible in widely available commercial software -- including Microsoft Excel -- thereby facilitating distributional evaluation and quantile computation. We also obtain a Gegenbauer separation-of-variables expansion for the associated harmonic extension and its threshold derivative, with coefficients in closed Beta-Gamma form, and derive sharp tail expansions with explicit leading constants and higher-order corrections.

A model is multicalibrated on a collection of group weights $G$ if it is calibrated -- i.e. unbiased even conditional on its prediction -- not just overall, but also after reweighting contexts by each $g \in G$. It is a useful property for many downstream applications and is a basic desideratum of trustworthy machine learning. Before this work, all predictors known to attain the minimax-optimal $\widetilde O(\varepsilon^{-3})$ sample complexity rate for $\varepsilon$-multicalibration were randomized, while deterministic predictors were known only with substantially worse sample complexity. Whether randomization is necessary for optimal sample complexity in multicalibration was explicitly asked by [CLNR26] and implicitly in several prior works. We resolve this open problem by giving a minimax-optimal multicalibration algorithm that outputs a deterministic predictor. We then generalize the algorithm to produce optimal deterministic predictors that satisfy outcome indistinguishability (OI) with respect to finite or finitely covered collections of tests. As an application, this also gives deterministic omnipredictors and panpredictors with optimal sample complexity, resolving open problems posed by [OKK25] and [BHHLZ25].

Standard MCMC diagnostics ($\hat{R}$, effective sample size, divergence counts) detect whether a chain has mixed, but not why it has not. We ask whether the centring/non-centring obstruction in hierarchical models has a geometric cause beyond the metric. The joint parameter space is a fiber bundle (hyperparameters the base, group-level parameters the fibers), and the Fisher information metric induces an Ehresmann connection $A = -G_{FF}^{-1}G_{BF}$; the natural hypothesis is that the obstruction is its curvature, felt by the sampler as holonomy. We prove this false. The connection is flat for any smooth hierarchical posterior, not only the Gaussian case, because its horizontal leaves are the level sets of the fiber score $\partial_α\log p$: there is no geometric obstruction above the metric. What remains is statistical, not geometric, and the flat connection identifies it as a single quantity: the conditional dependence of fiber on base, governed per group by the prior fraction $π_j$, the classical pooling factor. From it the framework recovers the established picture, that prior-dominated groups mix slowly and that the optimal per-group non-centring weight follows in closed form, and a simulation study separates this base-fiber coupling from the funnel, a distinct base-space pathology, by their opposite dependence on the hierarchical variance. A direct attribution test confirms that NUTS does not transport the fiber: the chain-level footprint is excess conditional autocorrelation in prior-dominated groups, exactly as $π_j$ predicts. Genuine, even rotational, curvature does appear, but only for connections built from a sampler's working metric (a fixed mass matrix), where holonomy re-enters as an algorithmic rather than geometric phenomenon. The prior-fraction diagnostic is distributed as the R package fibr, with the geometric methods as accompanying reproduction code.

Recent research on Doeblin coefficients has shed light on their usefulness as a multi-way generalization of the Dobrushin contraction coefficient for TV distance, in a separate vein from their classic role in the theory of Markov chain ergodicity. However, strong conditions, such as being bounded away from 0, are typically necessary for Doeblin coefficients to establish the existence of information contraction. Building on recently formulated concepts of nonlinear information contraction, we aim to propose a finer-grained Doeblin-based characterization of multi-way contraction behavior which yields non-vacuous contraction guarantees even for channels whose Doeblin coefficient is 0. To this end, we introduce the notion of a Doeblin curve -- a nonlinear function which quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through the course of our analysis, we develop a new variational characterization of Doeblin coefficients, present several properties of Doeblin curves, define several versions of power-constrained Doeblin curves, and derive upper and lower bounds using our aforementioned variational characterization. We then utilize these results in diverse areas, including generalization bounds for noisy iterative optimization, error bounds for reliable computation with noisy circuits, and differential privacy guarantees for online iterative algorithms. In particular, we extend results in these areas to broader domains or group settings, leveraging Doeblin curves to reveal finer-grained contraction phenomena than Doeblin coefficients.

Organizations and researchers show increasing interest in using large language models (LLMs) in place of human participants in A/B tests, in the hope of experimenting faster and at lower cost. We study when a treatment effect estimated on LLM outcomes can recover the effect that would have been measured on the human population of interest. Distributional equivalence between LLM and human outcomes would make any standard estimator valid but is unrealistic. We therefore develop a statistical framework that adapts surrogate endpoint theory to LLMs, showing that calibrating LLM outcomes to human outcomes identifies the average treatment effect under surrogacy and comparability conditions that are jointly weaker than distributional equivalence. We present a falsification test for surrogacy and a bound on the worst-case bias from limited overlap between the LLM and human samples. We further show that the stochasticity inherent to LLMs can weaken surrogacy for identification while also introducing bias and variance during estimation, but that using an average over multiple LLM draws per unit as the surrogate mitigates these issues. Simulations validate the results, and an empirical application to A/B tests on Upworthy headlines shows that raw LLM predictions recover only 39\% of the human treatment effect while nonparametric calibration closes the gap. A central takeaway is that A/B testing on LLMs yields correct results only by assumption, whereas A/B testing on humans is correct by design, and that the required assumptions are hardest to justify precisely where A/B testing on LLMs promises the greatest benefit. We discuss the role of LLM choice, prompting, and temperature as design variables, the compounded challenge posed by long-term outcomes, and how to size human pilot studies for validation.

We develop indexed Bellman information complexity, a representation-level theory of interactive decision making centered on information indices and reference histories. The representation strips away problem-specific syntax and retains only the ingredients needed for dynamic programming and information accounting, thereby unifying the earlier framework of indexed algorithmic information ratios (AIR). On the upper-bound side, regret is controlled by Bellman supersolutions or potential identities whose gradient bracket is paid for by indexed information. Upper-confidence-bound (UCB), estimation-to-decision/decision-estimation-coefficient (E2D/DEC), and adaptive-minimax-sampling or exploration-by-optimization (AMS/EBO) methods appear as three relaxations of this same identity. On the lower-bound side, the posterior-reference trajectory supplies both the information telescope and the ghost quantile of small-regret trajectories. The resulting critical radius in the lower bound is an effective-dimension-scale quantity, as in Fano and local-prior-mass lower bounds, rather than the constant radius of a two-point Le Cam argument. The examples show that DEC is best viewed as a one-step relaxation of indexed Bellman information complexity, not as a universally tight conversion mechanism. We illustrate the framework through several applications, with particular emphasis on kernel bandits. In this setting, the active action marginal provides a concrete basis for comparing UCB, E2D, and AMS/EBO.

The expected signature uniquely determines the law of a random rough path under a moment-growth condition, yet finite-sample bounds for estimating its truncations from a single long dependent trajectory remain unavailable. We study a strictly stationary stochastic process equipped with a geometric rough-path lift, observed in non-overlapping blocks of equally-spaced samples, and prove a non-asymptotic mean-squared error (MSE) bound for the block-averaging estimator of its truncated expected signature. Under moment and stationarity assumptions together with a direct covariance-decay condition on block signatures -- strictly weaker than $α$-mixing and applicable to long-range-dependent processes -- the error separates into a discretization term and a fluctuation term, with rates determined respectively by path regularity and dependence strength. A levelwise rough-factorial variance analysis keeps finite-truncation constants explicit and yields an optimal allocation rule under a fixed observation budget. We verify the assumptions for independent-coordinate fractional Ornstein--Uhlenbeck processes in three regimes: short-range (Hurst $1/4<H<1/2$), semimartingale ($H=1/2$), and long-range ($H>1/2$); in all three, the block-signature covariance is summable, so the fluctuation term decays at the same rate as in the independent-block case, even under long memory at $H>1/2$. Monte Carlo experiments show empirical slopes steeper than the guaranteed upper-bound rates.

We revisit size controllability results in Pötscher and Preinerstorfer (2025) concerning heteroskedasticity robust test statistics in regression models. For the special, but important, case of testing a single restriction (e.g., a zero restriction on a single coefficient), we povide a necessary and sufficient condition for size controllability, whereas the condition in Pötscher and Preinerstorfer (2025) is, in general, only sufficient (even in the case of testing a single restriction).

The article establishes a rigorous shift operator calculus for stationary time series modeling, addressing a certain gap in the literature. It provides proofs of existence and isometry for the transfer function operators $f(B)$ and $f(T)$ where $B$ is the bilateral shift operator and $T$ is the unilateral shift operator for different families of functions $f$. The article establishes convergence of the power series of $f(B)$ and $f(T)$ under the operator norm for the Wiener algebra $\mathbb{W}_+$, and convergence under strong operator topology for $f$ in $H^{\infty}$, based on the use of Abel sums. Based on this calculus, it unifies the notion of stationary process invertibility with the operator invertibility of the transfer function $f(T)$.

Standard approaches to quantum statistical inference rely on measurements that induce a collapse of the wave function, effectively consuming the quantum state to extract information. In this work, we investigate the fundamental limits of \emph{locally-gentle} quantum state certification, where the learning algorithm is constrained to perturb the state by at most $α$ in trace norm, thereby allowing for the reuse of samples. We analyze the hypothesis testing problem of distinguishing whether an unknown state $ρ$ is equal to a reference $ρ_0$ or $ε$-far from it. We derive the minimax sample complexity for this problem, quantifying the information-theoretic price of non-destructive measurements. Specifically, by constructing explicit measurement operators, we show that the constraint of $α$-gentleness imposes a sample size penalty of $\frac{d}{α^2}$, yielding a total sample complexity of $n = Θ(\frac{d^3}{ε^2 α^2})$. Our results clarify the trade-off between information extraction and state disturbance, and highlight deep connections between physical measurement constraints and privacy mechanisms in quantum learning. Crucially, we find that the sample size penalty incurred by enforcing $α$-gentleness scales linearly with the Hilbert-space dimension $d$ rather than the number of parameters $d^2-1$ typical for high-dimensional private estimation.

Predictions from statistical physics postulate that recovery of the communities in the Stochastic Block Model (SBM) with a fixed number $K$ of communities is possible in polynomial time above, and only above, the Kesten-Stigum (KS) threshold. This conjecture has given rise to a rich literature, proving that non-trivial community recovery is indeed possible in SBM above the KS threshold. Failure of low-degree polynomials (LDP) below the KS threshold was also proven, as long as $K\ll \sqrt{n}$, where $n$ is the number of nodes in the observed graph. When $K\geq \sqrt{n}$, Chin et al.(2025) recently proved that, in a \emph{sparse regime}, community recovery in polynomial time is possible below the KS threshold by counting non-backtracking paths. This breakthrough led them to postulate a new threshold for the many-communities regime $K\geq \sqrt{n}$. In this work, we provide evidence supporting their conjecture:\\ 1- We prove that, for \emph{any graph density}, LDP fail to recover communities below the threshold postulated by Chin et al.(2025) ;\\ 2- We prove that community recovery is possible in polynomial time above the postulated threshold, not only in the \emph{sparse regime} considered in Chin et al.~(2025), but also in \emph{moderately sparse regimes}, by counting occurrences of some specific motifs inspired by the LDP analysis.\\ In particular, counting self-avoiding paths of length $\log(n)$, which is closely related to spectral algorithms based on the Non-Backtracking operator, is optimal only in the sparse regime. More complex motifs based on the blow-up of a cycle must be considered in denser regimes.

In this paper, we develop the mathematical framework for filtering problems arising from biophysical applications where data is collected from confocal laser scanning microscopy recordings of the space-time evolution of intracellular wave dynamics of biophysical quantities. In these applications, signals are described by stochastic partial differential equations (SPDEs) and observations can be modelled as functionals of marked point processes whose intensities depend on the underlying signal. We derive both the unnormalized and normalized filtering equations for these systems, demonstrate the asymptotic consistency and approximations of finite dimensional observation schemes respectively partial observations. Our theoretical results are validated through extensive simulations using synthetic and real data. These findings contribute to a deeper understanding of filtering with point process observations and provide a robust framework for future research in this area.

Statistical literature provides pointwise limiting distributions of the nonparametric maximum likelihood estimator (NPMLE) in monotone binary regression for the two extremal cases: If the feature-label relation is strictly monotone and sufficiently smooth, it converges at a cube-root-$n$ rate with scaled Chernoff-type limiting distribution, and it converges at the parametric $\sqrt{n}$-rate if the underlying relation is flat. In this article, we provide the complete picture of the distributional metamorphosis of the NPMLE, revealing a new limiting distribution which provides a significantly better distributional approximation for small samples in case of a weak feature-label relationship. It is shown to continuously interpolate between the two extremal cases. The innovative way to determine this distribution is to generate it as a limit of the NPMLE in the newly introduced weak-feature-impact triangular array for a particular parameter-sample-size constellation. Moreover, the phase transition is likewise observed for the suitably rescaled $L^{1}$-error in this weak-feature-impact scenario. As a by-product, its limiting distribution for flat regression functions is obtained, which was unknown before. The proof develops a completely new strategy, notably not based on the switch relation. A novel type of local minimax lower bounds accompanies these results.

Piecewise-Deterministic Markov Processes (PDMPs) hold significant promise for sampling from complex probability distributions. However, their practical implementation is hindered by the need to compute model-specific bounds. Conversely, while Hamiltonian Monte Carlo (HMC) offers a generally efficient approach to sampling, its inability to adaptively tune step sizes impedes its performance when sampling complex distributions like funnels. To address these limitations, we introduce three innovative concepts: (a) a Metropolis-adjusted approximation for PDMP simulation that eliminates the need for explicit bounds without compromising the invariant measure, (b) an adaptive step size mechanism compatible with the Metropolis correction, and (c) a No U-Turn Sampler (NUTS)-inspired scheme for dynamically selecting path lengths in PDMPs. These three ideas can be seamlessly integrated into a single, `doubly-adaptive' PDMP sampler with favourable robustness and efficiency properties.

Recent advances in deep learning have highlighted the phenomenon of benign overfitting in overparameterized statistical models, sparking significant interest in understanding its foundations. Owing to its simplicity and practical relevance, the ordinary least squares (OLS) interpolator has become a key object of study for gaining theoretical insight into this phenomenon. While the properties of OLS are well understood in classical underparameterized settings, its behavior in the overparameterized regime -- unlike that of ridge regression or the lasso -- remains comparatively less explored. We contribute to this growing literature by deriving new algebraic and statistical results for the minimum $\ell_2$-norm OLS interpolator. In contrast to much of the existing work, which focuses on prediction risk, we center our analysis on parameter estimation and inference, which are fundamental for many statistics and causal inference applications. Specifically, we establish overparameterized analogues of (i) the leave-$k$-out formulas, (ii) the omitted variable bias formula, and (iii) the Frisch-Waugh-Lovell theorem. Under the Gauss-Markov model, we further extend the Gauss-Markov theorem and analyze variance estimation under homoskedasticity in the overparameterized setting. Collectively, these results provide a systematic framework for studying parameter estimation and inference in overparameterized linear models, offering a novel perspective on benign overfitting beyond its implications for prediction.

When estimating causal effects from observational studies, researchers often need to adjust for many covariates to deconfound the non-causal relationship between exposure and outcome, among which many covariates are discrete. The behavior of commonly used estimators in the presence of many discrete covariates is not well understood since their properties are often analyzed under structural assumptions including sparsity and smoothness, which do not apply in discrete settings. In this work, we study the estimation of causal effects in a model where the covariates required for confounding adjustment are discrete but high-dimensional, meaning the number of categories $d$ is comparable with or even larger than sample size $n$. Specifically, we show the mean squared error of commonly used regression, weighting and doubly robust estimators is bounded by $\frac{d^2}{n^2}+\frac{1}{n}$. We then prove the minimax lower bound for the average treatment effect is of order $\frac{d^2}{n^2 \log^2 n}+\frac{1}{n}$, which characterizes the fundamental difficulty of causal effect estimation in the high-dimensional discrete setting, and shows the estimators mentioned above are rate-optimal up to log-factors. We further consider additional structures that can be exploited, namely effect homogeneity and prior knowledge of the covariate distribution, and propose new estimators that enjoy faster convergence rates of order $\frac{d}{n^2} + \frac{1}{n}$, which achieve consistency in a broader regime. The results are illustrated empirically via simulation studies.

We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $\mathbb{X}$ equipped with a continuous function $f: \mathbb{X} \rightarrow \mathbb{R}$. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line $\mathbb{R}$. We then introduce a variant of the classic mapper graph of Singh et al.~(2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $(\mathbb{X}, f)$ when it is applied to points randomly sampled from a probability density function concentrated on $(\mathbb{X}, f)$. Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (2018), we first show that the mapper graph of $(\mathbb{X}, f)$, a constructible $\mathbb{R}$-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $(\mathbb{X},f)$ to the mapper of a super-level set of a probability density function concentrated on $(\mathbb{X}, f)$. Finally, building on the approach of Bobrowski et al.~(2017), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.