arXiv subjects

Aaditya Ramdas

Publications and source records attributed to Aaditya Ramdas.

At least 19 recordsLinked to original sources

arXiv2026

Distribution-free inference on the number of changepoints

Suppose we are given an ordered sequence of independent data whose distribution changes $K$ times at unknown locations, for some unknown $K \geq 0$. In this paper, we study the problem of performing distribution-free inference on $K$. First, we show an impossibility result: any distribution-free upper confidence bound on $K$ must be trivial and uninformative. Then, using conformal $p$-values, and under only the assumption that the data segments induced by the changepoints are exchangeable (within themselves) and mutually independent, we construct a finite-sample valid lower confidence bound on $K$, which we call the Conformal LOwer bound on Changepoint Count (CLOCC). We show that CLOCC is the only feasible way to provide a lower bound on $K$ under the stated assumptions, a property we refer to as its universality. We provide practical guidelines for choosing score functions that yield efficient and tight lower bounds. We evaluate CLOCC in several synthetic and real-data experiments, where it provides informative lower bounds on $K$, demonstrating its practical applicability.

stat.ML↗

arXiv2026

A complete characterization of sequential testability and change detectability in i.i.d. models

We give a necessary and sufficient condition for the existence of power-one sequential tests in an i.i.d. composite testing problem. A level-$\alpha$ test with power one against every alternative exists if and only if the alternatives are separated from the null by a countable family of finite-block events. We provide other equivalent conditions using randomized fixed-sample tests, bounded finite-block scores, e-processes, reduced-filtration test supermartingales, and a countable cover whose finite-block weak-$*$ closed convex hulls are positively separated in total variation. As a bonus, the constructive proof yields tests have pointwise expected sample size $O_Q(\log(1/\alpha))$. Exactly the same conditions also characterize i.i.d.\ change detectability under optional-horizon average-run-length control: for every $\eta>0$, they are equivalent to an alarm family $(T_\gamma)_{\gamma\ge1}$ satisfying $\Prob_{P^\infty}(T_\gamma\le\sigma)\le \E_{P^\infty}\sigma/\gamma$ for every null law and every stopping time $\sigma$. In fact, when these conditions hold, we can construct a single e-detector such that every null-law average run length lies between $\gamma$ and $(1+\eta)\gamma+1$, and having robust Lorden delay $O_Q(\log\gamma)$.

math.ST↗

arXiv2026

Conformal Prediction Through the Lens of Hypothesis Testing: Universality, Impossibility, and Optimality

The connections between conformal prediction and permutation tests are already widely-known in the literature. Some authors motivate conformal prediction by saying that it computes a permutation p-value for the hypothesis $H_0 : Y_{n+1} = y$, and then inverts this to form a prediction set for $Y_{n+1}$ (i.e., accepts all values $y$ into the prediction set for which the p-value is large). In this paper, we examine an alternative view, which is less well-known: we again cast conformal prediction via the inversion of a permutation test, but for the null of exchangeability of the joint distribution of the $n+1$ samples. This change in perspective, while simple, adheres more closely to traditional formalization in hypothesis testing, which offers several benefits. First, we use the duality between conformal sets and testing to show that foundational universality and impossibility results in the conformal prediction literature can be reproduced directly using classical hypothesis testing theory (due to Neyman, Lehmann, Scheff\'e, Kraft, Le Cam, and others). Furthermore, we show that an optimality result for conformal prediction can be derived using standard Neyman-Pearson theory: for any joint distribution of the covariates and response $X,Y$, and any sample size, the optimal method for prediction sets---which delivers the most efficient set among all methods with valid coverage for exchangeable distributions---is a conformal predictor whose score is the inverse conditional density of $Y|X$.

math.ST↗

arXiv2026

Gaussian-efficient testing by betting on the mean of bounded data

Given $[0,1]$-valued random variables $X_1,\dots,X_n$ such that $\mathbb{E}[X_i | X_1,\dots,X_{i-1}]= \mu$ for all $i$, we propose a new nonasymptotic confidence interval for $\mu$ that is obtained by inverting terminal e-values generated by a novel betting strategy. When the data are iid, its limiting width matches that of the central limit theorem (``Gaussian-efficient''), finally surpassing the inefficient limits of previous betting intervals. Our main conceptual advance involves designing betting fractions that track the conditional rejection probability of the most powerful terminal test in a limiting Gaussian experiment. When one predictable variance estimator is shared across candidate means, the deterministic inversion is an interval for every data sequence and its two endpoints can be found easily. The width can be improved further with external randomization. In simulations, our method yields the tightest intervals to date; for every distribution tested and all sufficiently large $n$, our deterministic version beats STaR-Bets and is competitive with Gaffke, while the randomized improvement beats both. It thus combines finite-sample validity under martingale dependence, easy endpoint computation, Gaussian-efficient inference for iid data, and excellent empirical performance. We also extend the construction and its efficiency theory to sampling without replacement, where it again achieves state-of-the-art empirical performance.

stat.ME↗

arXiv2026

Universality of e-detectors for ARL control

An e-detector for a pre-change class $\mathcal P$ is a nonnegative process $M$ such that $\mathbb E_P[M_\tau] \leq \mathbb E_P[\tau]$ for all stopping times $\tau$ and all $P \in \mathcal P$. Thresholding e-detectors controls the average run length (ARL): declaring a change at the first time $T_b$ when $M$ crosses $b$ ensures that $\inf_{P \in \mathcal P}\mathbb E_P[T] \geq b$. But e-detectors do substantially more than control the ARL; they also satisfy a \emph{optional-horizon inequality}: \[ P(T_b\leq\sigma)\leq \mathbb E_P[\sigma]/b \] for every data-dependent stopping time (monitoring horizon) $\sigma$ and $P\in \mathcal P$. In particular, every e-detector-based procedure obeys $P(T\leq t)\leq t/b$ at each fixed $t$, thus avoiding early false alarms. Remarkably, the converse also holds: every stopping time $T$ that satisfies the optional-horizon inequality must in fact arise from thresholding an e-detector. We also derive a universal representation of stopping times that satisfy (only) ARL control. These are represented by \emph{weak} e-detectors, that only require $\mathbb E_P[M_\tau] \leq \mathbb E_P[\tau]$ to hold at all threshold stopping times $T_b$. Appendices present universal representations for other (less common) change detection metrics.

math.ST↗

arXiv2026

Non-partitioned e-detectors for nonparametric sequential change detection

We study the problem of sequential change detection over a general class of probability distributions ($\mathcal P$), where both the pre-change and post-change distributions are unknown and belong to $\mathcal P$. We do not assume a pre-specified partition of $\mathcal P$ into pre- and post-change families. We propose a general class of sequential change detectors obtained by aggregating point-null e-processes over possible changepoints and taking an infimum over candidate no-change distributions. The weights in the aggregation scheme determine whether they attain average run length (ARL) control and probability-of-false-alarm (PFA) control. Under suitable assumptions, we prove that our methods achieve first-order asymptotically optimal detection delay. Concrete examples include sub-Gaussian and bounded mean changes, Gaussian mean changes with unknown variance, as well as changes in Markov transition matrices.

stat.ME↗

arXiv2026

Monte Carlo testing: non-asymptotic guarantees without joint exchangeability

In hypothesis testing, Monte Carlo tests are usually justified either by exact null simulation or by joint exchangeability of the observed data and its simulated copies. This leaves a gap for common computational procedures, such as parallel MCMC sampling initialized at the observed data, where each copy may be marginally null and even pairwise exchangeable with the observation, but the full collection is not jointly exchangeable. In such cases the usual empirical p-value can be invalid when the chain has not mixed, while exactly exchangeable constructions such as the Besag--Clifford hub-and-spoke sampler may suffer from high conditional Monte Carlo variability. We give finite-sample guarantees for this intermediate regime. If, under the null, the observed data $X$ and a copy $X'\sim P(\cdot\mid X)$ are conditionally i.i.d.\ given a latent variable, then for any prespecified statistic and any finite number $m$ of conditionally independent Monte Carlo copies, the resulting empirical p-value obeys $\mathbb P\{p_m\le \alpha\}\le 2\alpha$. This guarantee requires no mixing conditions and holds for any number of copies $m$, and it explains finite-sample oscillatory behavior in inference via MCMC sampling. In addition, we further show that the guarantee provides insights into inference problems arising in other settings, including inference on Bayesian models (recovering a classical result showing validity up to a factor of $2$ for posterior predictive p-values), and inference via balanced permutation tests.

stat.ME↗

arXiv2026

Gaffke's confidence interval for the mean of bounded data is inadmissible but asymptotically efficient

Given observations $\mathbf x=(x_1,\dots,x_n)$, Gaffke (2005) defined \[ K_n(\mathbf x)=\mathbb{P}_{\mathbf D}\!\left\{\sum_{i=1}^n x_iD_i\le 1\right\}, \qquad (D_0,D_1,\ldots,D_n)\sim\mathrm{Dirichlet}(1,\ldots,1), \] and conjectured that it is a $p$-value whenever the inputs are independent e-values. Recently, Vlassis and Thomas (2026) proved this conjecture. Inverting the tests for observations in $[0,1]$ gives the confidence interval studied by Learned-Miller and Thomas (2020), which reduces to Clopper--Pearson for Bernoulli data. We give a finite- and large-sample account of Gaffke's test and interval. First, for every $\mathbf x\in[0,\infty)^n$ and every elementary symmetric polynomial $e_k$, $ K_n(\mathbf x)e_k(\mathbf x)\le {n\choose k}, $ so the Gaffke $p$-value never larger than the SymPol $p$-value of Ming et al. (2026). However, Gaffke's p-value is inadmissible. For $n=2$, we construct a valid rule that is strictly smaller on mixed configurations and is the unique admissible rule that dominates $K_2$. A neutral-face extension proves inadmissibility of $K_n$ for every $n\ge2$. If one independent uniform random variable is allowed, there is an even simpler full-dimensional improvement: on the upper orthant, where $K_n(\mathbf x)=1/\prod_i x_i$, replace it by $U/\prod_i x_i$. The equal-tail Gaffke confidence interval $I_n$ is nevertheless first-order asymptotically efficient: for iid observations on $[0,1]$ with unknown variance $\sigma^2>0$, \[ \sqrt n\,\operatorname{Width}(I_n)\longrightarrow 2\sigma z_{1-\alpha/2}\qquad\text{almost surely}. \] Our simulations also find that, among a variety of bounded-mean intervals considered, the Gaffke interval is the shortest, including comparisons with a recent empirical Berry--Esseen procedure having the same first-order Gaussian target.

math.ST↗

arXiv2026

Strong duality for the GROW criterion

This paper presents general strong duality results when testing hypotheses by betting against them. A bet is an e-variable for a composite null hypothesis $\Pcal$: a nonnegative random variable $X$ whose expected value is at most one under every $P \in \mathcal P$. Following Kelly, Breiman, Cover, Shafer, and Grunwald et al. (2024), we study a natural minimax \emph{log-optimality} criterion: given a composite alternative $\Qcal$, we characterize the ``GROW value'' $\sup_{X} \inf_{Q} \E_{Q}[\log X]$. This paper generalizes the results of Larsson et al. (2025) from (arbitrary $\mathcal P$ and) simple $\mathcal Q$ to arbitrary $\mathcal Q$. We prove that there always exists a minimizing information-projection pair between the weak-$*$ closures of the convex hulls of arbitrary $\mathcal P$ and $\mathcal Q$, and show that the GROW value for \emph{bounded} e-variables always equals their relative entropy. We also prove a similarly general strong duality for the REGROW criterion with bounded e-variables and arbitrary bounded offsets. Under various assumptions our results extend to unbounded e-variables, and examples show that without any assumptions such extensions fail. Our results are analogous to those in Larsson et al. (2026), swapping tests for bounded e-variables, minimax risk for the GROW criterion, and total variation for relative entropy.

math.ST↗

arXiv2026

Bentkus-type asymptotic e-values

Asymptotic e-values are emerging as a powerful alternative to asymptotic p-values, particularly in post-hoc inference and multiple testing, where significance levels may be data-dependent. Existing asymptotic e-values, however, suffer from the ``missing factor,'' a scaling inefficiency resulting in overly conservative inference. Drawing on the framework of near-optimal concentration inequalities developed by Bentkus in the 2000s, we introduce Bentkus-type asymptotic e-values and prove that they successfully eliminate the missing factor. We also demonstrate both theoretically and empirically that Bentkus-type e-values consistently deliver sharper inference than existing alternatives, leading to tighter post-hoc confidence intervals and higher rejection rates in multiple testing procedures.

math.ST↗

arXiv2026

Distribution-free changepoint localization after sequential change detection

This paper introduces a distribution-free framework for constructing post-detection confidence sets for changepoints after stopping a sequential change detection procedure. It is well known that conformal test martingales can be used to sequentially detect changes in distribution, but by themselves provide no inference for the time at which a proclaimed change occurred. Past work on post-detection inference requires pre- and post-change classes of distributions to be known, but this paper accomplishes localization of the changepoint without any distributional assumptions. We establish finite-sample coverage guarantees (conditional on correct detection). We provide non-asymptotic bounds on the conditional expected size of the confidence sets. Under suitable asymptotic regimes, we prove that the conditional expected size of the confidence set remains uniformly bounded and demonstrate strong empirical performance on simulated and real data. To the best of our knowledge, this is the first general distribution-free framework for sequential changepoint localization with valid post-detection coverage.

stat.ML↗

arXiv2026

M-estimation with e-statistics

We present a theory of point estimation with e-statistics (e-values and e-processes) by introducing the "ME-estimator": the parameter that minimizes the corresponding e-statistic, or the evidence against it. Our approach is based on the intuitive idea of e-statistics as a measure of evidence and betting pay-off, and naturally generalizes the classical method of maximum likelihood estimation. First, we establish the consistency as well as the almost sure convergence rate for ME-estimators relating to the high-probability bounds on the size of the confidence set derived from thresholding the e-statistics, an approach that sets ME-estimators apart from traditional M-estimators. Second, we conduct classical M-estimator-style analysis on the consistency and asymptotic normality of ME-estimators in the bounded mean estimation setting, discussing the notion of efficiency (or lack thereof) from various choices of betting strategy. Our work brings e-statistics, a fundamental tool for inference and uncertainty quantification, to the space of estimation.

stat.ME↗

arXiv2026

Optimal Rates for Differentially Private Hypothesis Testing with E-values

E-values have attracted considerable interest in recent years as flexible tools for enabling anytime-valid and adaptive data analysis. Hypothesis testing is at the core of many of these applications, which can often involve private or sensitive data. In this work, we answer a simple but important question: given two distributions $\mathbb{P}$ and $\mathbb{Q}$, what is the maximum achievable e-power when testing $X\sim \mathbb{P}^n$ against $X\sim\mathbb{Q}^n$ with e-values that satisfy $\varepsilon$-differential privacy? We characterize the optimal rate for this problem and provide an algorithm which matches it exactly. In the sequential setting, when observations arrive one-by-one and the analyst chooses when to halt, we give matching upper and lower bounds on the stopping times of any private e-process. Numerical experiments confirm the practicality of our algorithms, which require less data than the recently proposed DP-SPRT across a range of sequential testing problems and privacy levels.

cs.CR↗

arXiv2026

Distribution-free root cause analysis

We study distribution-free root cause analysis in multi-stream data, where an evolving underlying system is observed through multiple data streams that may each undergo distributional changes at unknown timepoints. In such settings, the stream exhibiting the earliest change provides a natural starting point for investigating the underlying cause, which we refer to as the root-cause index. Leveraging conformal $p$-values, we propose a novel framework, Conformal Root Cause Analysis (CROC), which constructs finite-sample valid confidence sets for the root-cause index under minimal assumptions: the data streams are independent, and within each stream the pre- and post-change observations are sampled exchangeably from arbitrary and unknown distributions. We further establish a universality property, showing that any distribution-free method for root cause localization can be represented within the CROC framework. In addition, under mild regularity conditions and principled score design, our method yields asymptotically sharp confidence sets that efficiently isolate the root cause. We further extend CROC to efficiently handle cross-stream dependence when present. Extensive simulations demonstrate accurate localization of the root stream, supporting our theoretical guarantees.

stat.ME↗

arXiv2026

Intrinsic-dimension empirical Bernstein inequalities for bounded self-adjoint operators

Operator-valued concentration inequalities are foundational to the analysis of modern high-dimensional statistics and randomized algorithms. However, standard oracle bounds are frequently limited in practice: they require explicit a priori knowledge of the true variance, and often explicitly scale with the ambient dimension, rendering them vacuous for infinite-dimensional or heavily structured operators. Motivated by these challenges, we establish the first empirical Bennett and Bernstein inequalities for sums of independent, bounded, compact self-adjoint operators. Our fully data-driven bounds replace the unknown variance with an empirical estimate and rely strictly on the intrinsic dimension rather than the ambient dimension. This structural shift yields computable, dimension-free guarantees that are strictly sharper for non-isotropic random matrices and seamlessly extend to infinite-dimensional Hilbert spaces. We demonstrate that our empirical bounds achieve asymptotic sharpness with the best known oracle rates. Finally, as an independent byproduct, we derive novel empirical concentration guarantees for the intrinsic dimension itself.

math.ST↗

arXiv2026

Optimal sequential tests yield log-optimal e-processes

It has been recently shown that e-processes are sufficient for sequential testing in the following sense: every level-$\alpha$ sequential test can be obtained by thresholding an e-process at $1/\alpha$. However, in the above result, neither does the test have to be asymptotically optimal (in terms of stopping times) nor does the e-process have to be asymptotically log-optimal. It has separately been shown that asymptotically log-optimal e-processes yield asymptotically optimal sequential tests. In this paper, we prove the converse, arguably completing the story: it is possible to aggregate asymptotically optimal sequential tests into asymptotically log-optimal e-processes. This is accomplished by using a new class of WAIT e-processes: those that are Weighted Aggregates of Indicators of stopping Times that begin at zero, are nondecreasing and increase to infinity under the alternative at the optimal rate. Importantly, the paper discusses several nuances in the varied definitions of asymptotic (log-)optimality.

math.ST↗

arXiv2026

The optimal betting wealth growth rate

This paper characterizes the best possible rate of growth of wealth in a Kelly betting game when repeatedly betting against a general i.i.d. null hypothesis $\mathscr{P}$, but the data are drawn i.i.d from an arbitrary alternative $Q$. We prove that it equals $\lim_{n \to \infty}n^{-1}\inf_{P \in (\mathscr P)^n)^{\circ\circ}} \mathrm{KL}(Q^n,P)$, where ${\mathscr P}^n = \{P^n: P \in \mathscr{P}\}$ and $(\mathscr {P}^n)^{\circ\circ}$ is its bipolar, i.e., this rate is achievable and one cannot do better. This quantity is in general smaller than a more popular quantity in the literature, $\mathrm{KL}_{\inf}(Q,\mathscr{P}) := \inf_{P \in \mathscr P}\mathrm{KL}(Q,P)$. If $\mathrm{KL}_{\mathrm{inf}}(\cdot,\mathscr P)$ is weakly lowersemicontinuous (w.l.s.c.) at $Q$, we show that the two quantities are equal; in particular, this happens when $\mathscr P$ is weakly compact. For simple alternatives, we provide the first matching necessary and sufficient condition for when power-one sequential tests exist (without assumptions on $\mathscr P, Q$). We also derive the optimal worst-case growth rate against composite $\mathscr Q$. We emphasize that test supermartingales on reduced filtrations suffice for all i.i.d. testing problems, and more general e-processes are not required. We thus completely generalize the recent results of Larsson et al.~\cite{larsson2025numeraire} to the sequential setting.

math.ST↗

arXiv2026

Optimal e-variables under constraints

E-variables enable safe and anytime-valid inference, with log-optimal e-variables given by the likelihood ratio of the least favorable distributions (LFDs) when they exist in composite settings. While this unconstrained theory is well understood, one may need/wish to impose additional structural constraints, including differential privacy, quantization, boundedness, or moment restrictions. We show that under these constraints, log-optimal constrained e-variables can often be constructed by a simple \emph{optimize-then-constrain} principle: first compute the unconstrained log-optimal e-variable, then impose the constraint via an appropriate transformation. Thus, the constrained growth-rate optimization problem does not require solving for a different LFD pair; the constrained optimal solution is just a post-processing of the unconstrained optimal solution.

stat.ME↗