Papers

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for nonnegative data, with applications such as hyperspectral unmixing and topic modeling. NMF is a difficult problem in general (NP-hard), and its solutions are typically not unique. To address these two issues, additional constraints or assumptions are often used. In particular, separability assumes that the basis vectors in the NMF are equal to some columns of the input matrix. In that case, the problem is referred to as separable NMF (SNMF) and can be solved in polynomial-time with robustness guarantees, while identifying a unique solution. However, in real-world scenarios, due to noise or variability, multiple data points may lie near the basis vectors, which SNMF does not leverage. In this work, we rely on the smooth separability assumption, which assumes that each basis vector is close to multiple data points. We explore the properties of the corresponding problem, referred to as smooth SNMF (SSNMF), and examine how it relates to SNMF and orthogonal NMF. We then propose a convex model for SSNMF and show that it provably recovers the sought-after factors, even in the presence of noise. We finally adapt an existing fast gradient method to solve this convex model for SSNMF, and show that it compares favorably with state-of-the-art methods on both synthetic and hyperspectral datasets.

Testing intersections of null-hypotheses is an integral part of closed testing procedures for assessing multiple null-hypotheses under family-wise type 1 error control. Popular intersection tests such as the minimum p-value test are based on marginal p-values and are typically evaluated conservatively by disregarding simultaneous behavior of the marginal p-values. We consider a general purpose Wald type test for testing intersections of one-sided null-hypotheses. The test is constructed on the basis of the simultaneous asymptotic behavior of the p values. The simultaneous asymptotic behavior is derived via influence functions of estimators using the so-called stacking approach. In particular, this approach does not require added assumptions on simultaneous behavior to be valid. The resulting test is shown to have attractive power properties and thus forms the basis of a powerful closed testing procedure for testing multiple one-sided hypotheses under family-wise type 1 error control.

Classical estimators for ARIMA parameters (MLE, CSS, OLS) assume Gaussian innovations, an assumption frequently violated in financial and economic data exhibiting asymmetric distributions with heavy tails. We develop and validate the second-order polynomial maximization method (PMM2) for estimating ARIMA$(p,d,q)$ models with non-Gaussian innovations. PMM2 is a semiparametric technique that exploits higher-order moments and cumulants without requiring full distributional specification. Monte Carlo experiments (128,000 simulations) across sample sizes $N \in \{100, 200, 500, 1000\}$ and four innovation distributions demonstrate that PMM2 substantially outperforms classical methods for asymmetric innovations. For ARIMA(1,1,0) with $N=500$, relative efficiency reaches 1.58–1.90 for Gamma, lognormal, and $\chi^2(3)$ innovations (37–47% variance reduction). Under Gaussian innovations PMM2 matches OLS efficiency, avoiding the precision loss typical of robust estimators. The method delivers major gains for moderate asymmetry ($|\gamma_3| \geq 0.5$) and $N \geq 200$, with computational costs comparable to MLE. PMM2 provides an effective alternative for time series with asymmetric innovations typical of financial markets, macroeconomic indicators, and industrial measurements. Future extensions include seasonal SARIMA models, GARCH integration, and automatic order selection.

Mixtures of Gaussian Bayesian networks have previously been studied under full-covariance assumptions, where each mixture component has its own covariance matrix. We propose a mixture model with tied-covariance, in which all components share a common covariance matrix. Our main contribution is the derivation of its marginal likelihood, which remains analytic. Unlike in the full-covariance case, however, the marginal likelihood no longer factorizes into component-specific terms. We refer to the new likelihood as the BGe scoring metric for tied-covariance mixtures of Gaussian Bayesian networks. For model inference, we implement MCMC schemes combining structure MCMC with a fast Gibbs sampler for mixtures, and we empirically compare the tied- and full-covariance mixtures of Gaussian Bayesian networks on simulated and benchmark data.

When using Bayesian inference to support conservative software reliability assessments, it is useful to consider a collection of Bayesian inference problems, with the aim of determining the worst-case value (from this collection) for a posterior predictive probability that characterizes how reliable the software is. Using a Bernoulli process to model the occurrence of software failures, we explicitly determine (from collections of Bayesian inference problems) worst-case posterior predictive probabilities of the software operating without failure in the future. We deduce asymptotic properties of these conservative posterior probabilities and their priors, and illustrate how to use these results in assessments of safety-critical software. This work extends robust Bayesian inference results and so-called conservative Bayesian inference methods.

Fairness concerns are increasingly critical as machine learning models are deployed in high-stakes applications. While existing fairness-aware methods typically intervene at the model level, they often suffer from high computational costs, limited scalability, and poor generalization. To address these challenges, we propose a Bayesian data selection framework that ensures fairness by aligning group-specific posterior distributions of model parameters and sample weights with a shared central distribution. Our framework supports flexible alignment via various distributional discrepancy measures, including Wasserstein distance, maximum mean discrepancy, and $f$-divergence, allowing geometry-aware control without imposing explicit fairness constraints. This data-centric approach mitigates group-specific biases in training data and improves fairness in downstream tasks, with theoretical guarantees. Experiments on benchmark datasets show that our method consistently outperforms existing data selection and model-based fairness methods in both fairness and accuracy.

The World Development Indicators (WDI) database provides a wide range of global development data, maintained and published by the World Bank. Our wdiexplorer package offers a comprehensive workflow that sources WDI data via the WDI R package, prepares and explores country-level panel data of the WDI through computational functions to calculate diagnostic metrics and visualise the outputs. By leveraging the functionalities of wdiexplorer package, users can efficiently explore any indicator dataset of the WDI, compute diagnostic indices, and visualise the metrics by incorporating the pre-defined grouping structures to identify patterns, outliers, and other interesting features of temporal behaviours. This paper presents the wdiexplorer package, demonstrates its functionalities using the WDI: PM$_{2.5}$ air pollution dataset, and discusses the observed patterns and outliers across countries and within groups of country-level panel data.

We present an algebraic approach to evolutionary accumulation modelling (EvAM). EvAM is concerned with learning and predicting the order in which evolutionary features accumulate over time. Our approach is complementary to the more common optimisation-based inference methods used in this field. Namely, we first use the natural underlying polynomial structure of the evolutionary process to define a semi-algebraic set of candidate parameters consistent with a given data set before maximising the likelihood function. We consider explicit examples and show that this approach is compatible with the solutions given by various statistical evolutionary accumulation models. Furthermore, we discuss the additional information of our algebraic model relative to these models.

Bayesian inference, while foundational to probabilistic reasoning, is often hampered by the computational intractability of posterior distributions, particularly through the challenging evidence integral. Conventional approaches like Markov Chain Monte Carlo (MCMC) and Variational Inference (VI) face significant scalability and efficiency limitations. This paper introduces a novel, unifying framework for fast Bayesian updates by leveraging harmonic analysis. We demonstrate that representing the prior and likelihood in a suitable orthogonal basis transforms the Bayesian update rule into a spectral convolution. Specifically, the Fourier coefficients of the posterior are shown to be the normalized convolution of the prior and likelihood coefficients. To achieve computational feasibility, we introduce a spectral truncation scheme, which, for smooth functions, yields an exceptionally accurate finite-dimensional approximation and reduces the update to a circular convolution. This formulation allows us to exploit the Fast Fourier Transform (FFT), resulting in a deterministic algorithm with O(N log N) complexity – a substantial improvement over the O(N^2) cost of naive methods. We establish rigorous mathematical criteria for the applicability of our method, linking its efficiency to the smoothness and spectral decay of the involved distributions. The presented work offers a paradigm shift, connecting Bayesian computation to signal processing and opening avenues for real-time, sequential inference in a wide class of problems.

Ordinal categorical data are routinely encountered in many practical applications. When the primary goal is to construct a regression model for ordinal outcomes, cumulative link models represent one of the most popular choices to link the cumulative probabilities of the response with a set of covariates through a parsimonious linear predictor, shared across response categories. As the number of observations grows, standard sampling algorithms for Bayesian inference scale poorly, making posterior computation increasingly challenging for large datasets. In this article, we propose three scalable algorithms for approximating the posterior distribution of the regression coefficients in cumulative probit models relying on Variational Bayes and Expectation Propagation. We compare the proposed approaches with inference based on Markov Chain Monte Carlo, demonstrating superior computational performance and remarkable accuracy. Finally, we illustrate the utility of the proposed algorithms on a challenging case study to investigate the structure of a criminal network.

Can classical game-theoretic frameworks be extended to capture the bounded rationality and causal reasoning of AI agents? We investigate this question by extending Causal Normal Form Games (CNFGs) to sequential settings, introducing Sequential Causal Multi-Agent Systems (S-CMAS) that incorporate Pearl's Causal Hierarchy across leader-follower interactions. While theoretically elegant – we prove PSPACE-completeness, develop equilibrium refinements, and establish connections to signaling theory – our comprehensive empirical investigation reveals a critical limitation: S-CNE provides zero welfare improvement over classical Stackelberg equilibrium across all tested scenarios. Through 50+ Monte Carlo simulations and hand-crafted synthetic examples, we demonstrate that backward induction with rational best-response eliminates any strategic advantage from causal layer distinctions. We construct a theoretical example illustrating conditions where benefits could emerge ($\epsilon$-rational satisficing followers), though implementation confirms that even relaxed rationality assumptions prove insufficient when good instincts align with optimal play. This negative result provides valuable insight: classical game-theoretic extensions grounded in rational choice are fundamentally incompatible with causal reasoning advantages, motivating new theoretical frameworks beyond standard Nash equilibrium for agentic AI.

We establish finite-sample error bounds in expectation for transformed Fréchet means in Hadamard spaces under minimal assumptions. Transformed Fréchet means provide a unifying framework encompassing classical and robust notions of central tendency in metric spaces. Instead of minimizing squared distances as for the classical 2-Fréchet mean, we consider transformations of the distance that are nondecreasing, convex, and have a concave derivative. This class spans a continuum between median and classical mean. It includes the Fréchet median, power Fréchet means, and the (pseudo-)Huber mean, among others. We obtain the parametric rate of convergence under fewer than two moments, and a subclass of estimators exhibits a breakdown point of 1/2. Our results apply in general Hadamard spaces—including infinite dimensional Hilbert spaces and nonpositively curved geometries—and yield new insights even in Euclidean settings.

The paper considers two different designs; a two-arm superiority cluster randomised controlled trial (RCT) with a continuous outcome, and a twoarm superiority cluster RCT with a binary outcome. From a Bayesian perspective, for the analysis of the trial we use a (generalised) linear mixed effects model. We summarise the inference for the treatment effect for a cluster RCT based on the posterior distribution. Based on this inference we use assurance to choose the sample size. We consider and compare two different methods for the inference: Markov Chain Monte Carlo (MCMC) and Integrated Nested Laplace Approximations (INLA), and consider their implications for the assurance. We consider the Specialist Pre-hospital redirection for ischemic stroke thrombectomy (SPEEDY) trial, an RCT which has co-primary outcomes of thrombectomy rate and time to thrombectomy, as a case study for the developed Bayesian RCT designs. We demonstrate our novel approach to the sample size calculation using assurance on the SPEEDY trial, based on the results of a formal prior elicitation exercise with two clinical experts. The paper considers a range of different scenarios for cluster RCTs to evaluate INLA and MCMC, to determine when each inference scheme should be used, balancing the computational cost in terms of speed and accuracy. We make recommendations for when each should be used.

The double descent (DD) paradox, where over-parameterized models see generalization improve past the interpolation point, remains largely unexplored in the non-stationary domain of Deep Reinforcement Learning (DRL). We present preliminary evidence that DD exists in model-free DRL, investigating it systematically across varying model capacity using the Actor-Critic framework. We rely on an information-theoretic metric, Policy Entropy, to measure policy uncertainty throughout training. Preliminary results show a clear epoch-wise DD curve; the policy's entrance into the second descent region correlates with a sustained, significant reduction in Policy Entropy. This entropic decay suggests that over-parameterization acts as an implicit regularizer, guiding the policy towards robust, flatter minima in the loss landscape. These findings establish DD as a factor in DRL and provide an information-based mechanism for designing agents that are more general, transferable, and robust.

We study the problem of detecting and localizing multiple changes in the mean parameter of a Banach space-valued time series. The goal is to construct a collection of narrow confidence intervals, each containing at least one (or exactly one) change, with globally controlled error probability. Our approach relies on a new class of weighted scan statistics, called Hölder-type statistics, which allow a smooth trade-off between efficiency (enabling the detection of closely spaced, small changes) and robustness (against heavier tails and stronger dependence). For Gaussian noise, maximum weighting can be applied, leading to a generalization of optimality results known for scalar, independent data. Even for scalar time series, our approach is advantageous, as it accommodates broad classes of dependency structures and non-stationarity. Its primary advantage, however, lies in its applicability to functional time series, where few methods exist and established procedures impose strong restrictions on the spacing and magnitude of changes. We obtain general results by employing new Gaussian approximations for the partial sum process in Hölder spaces. As an application of our general theory, we consider the detection of distributional changes in a data panel. The finite-sample properties and applications to financial datasets further highlight the merits of our method.

Irregularly sampled time series (ISTS), characterized by non-uniform time intervals with natural missingness, are prevalent in real-world applications. Existing approaches for ISTS modeling primarily rely on observed values to impute unobserved ones or infer latent dynamics. However, these methods overlook a critical source of learning signal: the reconstruction error inherently produced during model training. Such error implicitly reflects how well a model captures the underlying data structure and can serve as an informative proxy for unobserved values. To exploit this insight, we propose iTimER, a simple yet effective self-supervised pre-training framework for ISTS representation learning. iTimER models the distribution of reconstruction errors over observed values and generates pseudo-observations for unobserved timestamps through a mixup strategy between sampled errors and the last available observations. This transforms unobserved timestamps into noise-aware training targets, enabling meaningful reconstruction signals. A Wasserstein metric aligns reconstruction error distributions between observed and pseudo-observed regions, while a contrastive learning objective enhances the discriminability of learned representations. Extensive experiments on classification, interpolation, and forecasting tasks demonstrate that iTimER consistently outperforms state-of-the-art methods under the ISTS setting.

Two-phase sampling offers a practical way to collect costly confounders only in a subsample while retaining inexpensive information for a larger cohort. In observational causal studies, however, phase-2 selection can distort estimation of population causal effects if the sampling mechanism is ignored, and available phase-1 information may also be exploited to improve efficiency. Yet efficiency theory for causal estimands under such designs remains limited, particularly beyond the average treatment effect. In this paper, we derive the semiparametric efficiency bound for a class of propensity-score-weighted average treatment effects, which includes the average treatment effect, effects among treated and untreated populations, and the overlap effect, under two-phase sampling. In addition to straightforward weighting estimators based on the known sampling probabilities, we propose an enriched doubly robust estimator that attains the efficiency bound when all nuisance functions are consistently estimated. In particular, under outcome-dependent sampling, substantial efficiency gains can arise in some settings by appropriately incorporating phase-1 information. We further conduct extensive simulation studies, varying the choice of phase-1 variables and sampling schemes, to characterize when and to what extent leveraging phase-1 information leads to efficiency gains.

As the right to be forgotten becomes legislated worldwide, machine unlearning mechanisms have emerged to efficiently update models for data deletion and enhance user privacy protection. However, existing machine unlearning algorithms frequently neglect the fact that different data points may contribute unequally to model performance (i.e., heterogeneous data values). Treat them equally in machine unlearning procedure can potentially degrading the performance of updated models. To address this limitation, we propose Data Value-Weighted Unlearning (DVWU), a general unlearning framework that accounts for data value heterogeneity into the unlearning process. Specifically, we design a weighting strategy based on data values, which are then integrated into the unlearning procedure to enable differentiated unlearning for data points with varying utility to the model. The DVWU framework can be broadly adapted to various existing machine unlearning methods. We use the one-step Newton update as an example for implementation, developing both output and objective perturbation algorithms to achieve certified unlearning. Experiments on both synthetic and real-world datasets demonstrate that our methods achieve superior predictive performance and robustness compared to conventional unlearning approaches. We further show the extensibility of our framework on gradient ascent method by incorporating the proposed weighting strategy into the gradient terms, highlighting the adaptability of DVWU for broader gradient-based deep unlearning methods.

Robust causal discovery from observational data under imperfect prior knowledge remains a significant and largely unresolved challenge. Existing methods typically presuppose perfect priors or can only handle specific, pre-identified error types. And their performance degrades substantially when confronted with flawed constraints of unknown location and type. This decline arises because most of them rely on inflexible and biased thresholding strategies that may conflict with the data distribution. To overcome these limitations, we propose to harmonizes knowledge and data through prior alignment and conflict resolution. First, we assess the credibility of imperfect structural constraints through a surrogate model, which then guides a sparse penalization term measuring the loss between the learned and constrained adjacency matrices. We theoretically prove that, under ideal assumption, the knowledge-driven objective aligns with the data-driven objective. Furthermore, to resolve conflicts when this assumption is violated, we introduce a multi-task learning framework optimized via multi-gradient descent, jointly minimizing both objectives. Our proposed method is robust to both linear and nonlinear settings. Extensive experiments, conducted under diverse noise conditions and structural equation model types, demonstrate the effectiveness and efficiency of our method under imperfect structural constraints.

Higher criticism is a large-scale testing procedure that can attain the optimal detection boundary for sparse and faint signals. However, there has been a lack of knowledge in most existing works about its asymptotic distribution for more realistic settings other than the independent Gaussian assumption while maintaining the power performance as much as possible. In this paper, we develop a unified framework to analyze the asymptotic distributions of the higher criticism statistic and the more general multi-level thresholding statistic when the individual test statistics are dependent $t$-statistics under a finite ($2+\delta$)-th moment condition, $0<\delta\leq1$. The key idea is to approximate the global test statistic by the supremum of an empirical process indexed by a normalized class of indicator or thresholding functions, respectively. A new Gaussian approximation theorem for suprema of empirical processes with dependent observations is established to derive the explicit asymptotic distributions.

The widespread adoption of the maximum mean discrepancy (MMD) in goodness-of-fit testing has spurred extensive research on its statistical performance. However, recent studies indicate that the inherent structure of MMD may constrain its ability to distinguish between distributions, leaving room for improvement. Regularization techniques have the potential to overcome this limitation by refining the discrepancy measure. In this paper, we introduce a family of regularized kernel-based discrepancy measures constructed via spectral filtering. Our framework can be regarded as a natural generalization of prior studies, removing restrictive assumptions on both kernel functions and filter functions, thereby broadening the methodological scope and the theoretical inclusiveness. We establish non-asymptotic guarantees showing that the resulting tests achieve valid Type~I error control and enhanced power performance. Numerical experiments are conducted to demonstrate the broader generality and competitive performance of the proposed tests compared with existing methods.

Random forests are a statistical learning technique that use bootstrap aggregation to average high-variance and low-bias trees. Improvements to random forests, such as applying Lasso regression to the tree predictions, have been proposed in order to reduce model bias. However, these changes can sometimes degrade performance (e.g., an increase in mean squared error). In this paper, we show in theory that the relative performance of these two methods, standard and Lasso-weighted random forests, depends on the signal-to-noise ratio. We further propose a unified framework to combine random forests and Lasso selection by applying adaptive weighting and show mathematically that it can strictly outperform the other two methods. We compare the three methods through simulation, including bias-variance decomposition, error estimates evaluation, and variable importance analysis. We also show the versatility of our method by applications to a variety of real-world datasets.

We study the estimation of the ATE in randomized controlled trials under a dynamically evolving interference structure. This setting arises in applications such as ride-sharing, where drivers move over time, and social networks, where connections continuously form and dissolve. In particular, we focus on scenarios where outcomes exhibit spatio-temporal interference driven by a sequence of random interference graphs that evolve independently of the treatment assignment. Loosely, our main result states that a truncated Horvitz-Thompson estimator achieves an MSE that vanishes linearly in the number of spatial and time blocks, times a factor that measures the average complexity of the interference graphs. As a key technical contribution that contrasts the static setting we present a fine-grained covariance bound for each pair of space-time points that decays exponentially with the time elapsed since their last “interaction”. Our results can be applied to many concrete settings and lead to simplified bounds, including where the interference graphs (i) are induced by moving points in a metric space, or (ii) follow a dynamic Erdos-Renyi model, where each edge is created or removed independently in each time period.

Conventional topology learning methods for dynamical networks become inapplicable to processes exhibiting low-rank characteristics. To address this, we propose the low rank dynamical network model which ensures identifiability. By employing causal Wiener filtering, we establish a necessary and sufficient condition that links the sparsity pattern of the filter to conditional Granger causality. Building on this theoretical result, we develop a consistent method for estimating all network edges. Simulation results demonstrate the parsimony of the proposed framework and consistency of the topology estimation approach.

Wasserstein gradient and Hamiltonian flows have emerged as essential tools for modeling complex dynamics in the natural sciences, providing a unifying geometric formulation of many partial differential equations (PDEs) and finding applications in fields ranging from optimal transport to quantum mechanics and information geometry. Despite their significance, the inverse identification of potential functions and interaction kernels underlying these flows remains relatively unexplored. In this work, we tackle this challenge by addressing the inverse problem of simultaneously recovering the potential function and interaction kernel from discretized observations of the density flow. We formulate the problem as an optimization task that minimizes a loss function specifically designed to enforce the underlying variational structure of Wasserstein flows, ensuring consistency with the geometric properties of the density manifold. Our framework employs a kernel-based operator approach using the associated {reproducing kernel Hilbert space (RKHS)}, which provides a closed-form representation of the unknown components. Furthermore, we conduct a comprehensive error analysis, providing convergence rates under adaptive regularization parameters as the temporal and spatial discretization mesh sizes tend to zero. Moreover, a stability analysis is presented to bridge the gap between discrete trajectory data and continuous-time flow dynamics for the Wasserstein Hamiltonian flow. {Finally, numerical experiments on both Wasserstein gradient and Hamiltonian flows demonstrate accurate and robust recovery from discrete density observations, while comparisons with sparse-learning approaches illustrate the strong dependence of sparse recovery on the choice of dictionary.}

We study estimation of the average treatment effect (ATE) from a single network in observational settings with interference. The weak cross-unit dependence is modeled via an endogenous peer-effect (network autoregressive) term that induces distance-decaying network dependence, relaxing the common finite-order interference to infinite interference. We propose a targeted minimum loss estimation (TMLE) procedure that removes plug-in bias from an initial estimator. The targeting step yields an adjustment direction that incorporates the network autoregressive structure and assigns heterogeneous, network-dependent weights to units. We find that the asymptotic leading term related to the covariates $\mathbf{X}_i$ can be formulated into a $V$-statistic whose order diverges with the network degrees. A novel limit theory is developed to establish the asymptotic normality under such complex network dependent scenarios. We show that our method can achieve smaller asymptotic variance than existing methods when $\mathbf{X}_i$ is i.i.d. generated and estimated with empirical distribution, and provide theoretical guarantees for estimating the variance. Extensive numerical studies and a live-streaming data analysis are presented to illustrate the advantages of the proposed method.

The rapid adoption of large language models (LLMs), such as GPT-4 and Claude 3.5, underscores the need to distinguish LLM-generated text from human-written content to mitigate the spread of misinformation and misuse in education. One promising approach to address this issue is the watermark technique, which embeds subtle statistical signals into LLM-generated text to enable reliable identification. In this paper, we first generalize the likelihood-based LLM detection method of a previous study by introducing a flexible weighted formulation, and further adapt this approach to the inverse transform sampling method. Moving beyond watermark detection, we extend this adaptive detection strategy to tackle the more challenging problem of segmenting a given text into watermarked and non-watermarked substrings. In contrast to the approach in a previous study, which relies on accurate estimation of next-token probabilities that are highly sensitive to prompt estimation, our proposed framework removes the need for precise prompt estimation. Extensive numerical experiments demonstrate that the proposed methodology is both effective and robust in accurately segmenting texts containing a mixture of watermarked and non-watermarked content.

We consider the problem of transfer learning in Neyman-Pearson classification, where the objective is to minimize the error w.r.t. a distribution $\mu_1$, subject to the constraint that the error w.r.t. a distribution $\mu_0$ remains below a prescribed threshold. While transfer learning has been extensively studied in traditional classification, transfer learning in imbalanced classification such as Neyman-Pearson classification has received much less attention. This setting poses unique challenges, as both types of errors must be simultaneously controlled. Existing works address only the case of distribution shift in $\mu_1$, whereas in many practical scenarios shifts may occur in both $\mu_0$ and $\mu_1$. We derive an adaptive procedure that not only guarantees improved Type-I and Type-II errors when the source is informative, but also automatically adapt to situations where the source is uninformative, thereby avoiding negative transfer. In addition to such statistical guarantees, the procedures is efficient, as shown via complementary computational guarantees.

Uncertainty quantification (UQ) for adaptively collected data, such as that coming from adaptive experiments, bandits, or reinforcement learning, is necessary for critical elements of data collection such as ensuring safety and conducting after-study inference. The data's adaptivity creates significant challenges for frequentist UQ, yet Bayesian UQ remains the same as if the data were independent and identically distributed (i.i.d.), making it an appealing and commonly used approach. Bayesian UQ requires the (correct) specification of a prior distribution while frequentist UQ does not, but for i.i.d. data the celebrated Bernstein-von Mises theorem shows that as the sample size grows, the prior 'washes out' and Bayesian UQ becomes frequentist-valid, implying that the choice of prior need not be a major impediment to Bayesian UQ as it makes no difference asymptotically. This paper for the first time extends the Bernstein-von Mises theorem to adaptively collected data, proving asymptotic equivalence between Bayesian UQ and Wald-type frequentist UQ in this challenging setting. Our result showing this asymptotic agreement does not require the standard stability condition required by works studying validity of Wald-type frequentist UQ; in cases where stability is satisfied, our results combined with these prior studies of frequentist UQ imply frequentist validity of Bayesian UQ. Counterintuitively however, they also provide a negative result that Bayesian UQ is not asymptotically frequentist valid when stability fails, despite the fact that the prior washes out and Bayesian UQ asymptotically matches standard Wald-type frequentist UQ. We empirically validate our theory (positive and negative) via a range of simulations.

Understanding the relationship between brain activity and behavior is a central goal of neuroscience. Despite significant advances, a fundamental dichotomy persists: neural activity manifests as both discrete spikes of individual neurons and collective waves of populations. Both neural codes correlate with behavior, yet correlation alone cannot determine whether waves exert a causal influence or merely reflect spiking dynamics without causal efficacy. According to the Causal Hierarchy Theorem, no amount of observational data–however extensive–can settle this question; causal conclusions require explicit structural assumptions or careful experiment designs that directly correspond to the causal effect of interest. We develop a formal framework that makes this limitation precise and constructive. Formalizing epiphenomenality via the invariance of interventional distributions in Structural Causal Models (SCMs), we derive a certificate of sufficiency from Pearl's do-calculus that specifies when variables can be removed from the model without loss of causal explainability and clarifies how interventions should be interpreted under different causal structures of spike-wave duality. The purpose of this work is not to resolve the spike-wave debate, but to reformulate it. We shift the problem from asking which signal matters most to asking under what conditions any signal can be shown to matter at all. This reframing distinguishes prediction from explanation and offers neuroscience a principled route for deciding when waves belong to mechanism and when they constitute a byproduct of underlying coordination