Papers

We introduce a principled probabilistic framework for reward-guided decoding in large language models, addressing the limitations of standard decoding methods that optimize token-level likelihood rather than sequence-level quality. Our method defines a reward-augmented target distribution over complete sequences by combining model transition probabilities with prefix-dependent reward potentials. Importantly, the approach is training-free: it leaves model weights unchanged and instead modifies the inference distribution via reward potentials, with all gains arising purely from inference-time sampling. To sample from this distribution, we develop Sequential Monte Carlo algorithms, including a computationally efficient prefix-only variant and a lookahead variant whose intermediate targets match the exact marginals of the full sequence distribution. The framework also integrates resample-move updates with Metropolis-Hastings rejuvenation and supports block-wise generation, subsuming common decoding strategies such as temperature sampling and power-tempered objectives. Empirical results across three 7B models show significant gains. On code generation (HumanEval), our method improves base performance by up to 54.9% and surpasses the strongest sampling baselines by 9.1%-15.3%. On mathematical reasoning (MATH500), it achieves gains of up to 8.8%. Notably, it reaches 87.8% on HumanEval and 78.4% on MATH500 with Qwen2.5-7B, consistently outperforming the reinforcement learning method GRPO.

Group synchronization is a fundamental task involving the recovery of group elements from pairwise measurements. For orthogonal group synchronization, the most common approach reformulates the problem as a constrained nonconvex optimization and solves it using projection-based methods, such as the generalized power method. However, these methods rely on exact SVD or QR decompositions in each iteration, which are computationally expensive and become a bottleneck for large-scale problems. In this paper, we propose a Newton-Schulz-based Riemannian Gradient Scheme (NS-RGS) for orthogonal group synchronization that significantly reduces computational cost by replacing the SVD or QR step with the Newton-Schulz iteration. This approach leverages efficient matrix multiplications and aligns perfectly with modern GPU/TPU architectures. By employing a refined leave-one-out analysis, we overcome the challenge arising from statistical dependencies, and establish that NS-RGS with spectral initialization achieves linear convergence to the target solution up to near-optimal statistical noise levels. Experiments on synthetic data and real-world global alignment tasks demonstrate that NS-RGS attains accuracy comparable to state-of-the-art methods such as the generalized power method, while achieving nearly a 2$\times$ speedup.

Protecting individual privacy is essential across research domains, from socio-economic surveys to big-tech user data. This need is particularly acute in healthcare, where analyses often involve sensitive patient information. A typical example is comparing treatment efficacy across hospitals or ensuring consistency in diagnostic laboratory calibrations, both requiring privacy-preserving statistical procedures. However, standard equivalence testing procedures for differences in proportions or means, commonly used to assess average equivalence, can inadvertently disclose sensitive information. To address this problem, we develop differentially private equivalence testing procedures that rely on simulation-based calibration, as the finite-sample distribution is analytically intractable. Our approach introduces a unified framework, termed DP-TOST, for conducting differentially private equivalence testing of both means and proportions. Through numerical simulations and real-world applications, we demonstrate that the proposed method maintains type-I error control at the nominal level and achieves power comparable to its non-private counterpart as the privacy budget and/or sample size increases, while ensuring strong privacy guarantees. These findings establish a reliable and practical framework for privacy-preserving equivalence testing in high-stakes fields such as healthcare, among others.

We study stochastic convex optimization (SCO) with heavy-tailed gradients under pure $\varepsilon$-differential privacy (DP). Instead of assuming a bound on the worst-case Lipschitz parameter of the loss, we assume only a bounded $k$-th moment. This assumption allows for unbounded, heavy-tailed stochastic gradient distributions, and can yield sharper excess risk bounds. Prior work characterized the minimax optimal rate for $\rho$-zero-concentrated DP SCO up to logarithmic factors in this setting, but the pure $\varepsilon$-DP case has remained open. We characterize the minimax optimal excess-risk rate for pure $\varepsilon$-DP heavy-tailed SCO up to logarithmic factors. Our algorithm achieves this rate in polynomial time with high probability. Moreover, it runs in deterministic polynomial time when the worst-case Lipschitz parameter is polynomially bounded. For important structured problem classes – including hinge/ReLU-type and absolute-value losses on Euclidean balls, ellipsoids, and polytopes – we achieve deterministic polynomial time even when the worst-case Lipschitz parameter is infinite. Our approach is based on a novel framework for privately optimizing Lipschitz extensions of the empirical loss. We complement our upper bound with a nearly matching high-probability lower bound.

Most methods for learning with noisy labels require privileged knowledge such as noise transition matrices, clean subsets or pretrained feature extractors, resources typically unavailable when robustness is most needed. We propose Conformal Margin Risk Minimization (CMRM), a plug-and-play envelope framework that improves any classification loss under label noise by adding a single quantile-calibrated regularization term, with no privileged knowledge or training pipeline modification. CMRM measures the confidence margin between the observed label and competing labels, and thresholds it with a conformal quantile estimated per batch to focus training on high-margin samples while suppressing likely mislabeled ones. We derive a learning bound for CMRM under arbitrary label noise requiring only mild regularity of the margin distribution. Across five base methods and six benchmarks with synthetic and real-world noise, CMRM consistently improves accuracy (up to +3.39%), reduces conformal prediction set size (up to -20.44%) and does not hurt under 0% noise, showing that CMRM captures a method-agnostic uncertainty signal that existing mechanisms did not exploit.

Machine learning predictors are rarely deployed on data that match the data they were calibrated on. Conformal prediction and its risk-control generalization promise distribution-free guarantees at deployment, but only under exchangeability or a covariate shift whose likelihood ratio is known exactly – and the estimated-ratio case, the one that occurs in practice, is explicitly open, out of reach of the Bayesian construction available under exchangeability. We reach it from a different starting point: pushing a Dirichlet process prior on the calibration distribution through the likelihood ratio yields the exact Bayesian posterior over the deployed risk. That posterior is exact given the weight function, which at deployment is itself estimated, so we further prove finite-sample upper and lower bounds on the risk the selected threshold actually incurs, in terms of the weight-estimation error and the conditional shift. On synthetic and real covariate shifts, in regression and classification, the realized failure rate then stays close to target where shift-blind selection fails, and what deployment receives is a posterior over the risk of every threshold rather than a single number.

Statistical inference for stochastic block models typically relies on the spectrum of the normalized adjacency matrix $\A^*$. In practice, the true probability matrix $\mathbf{B}$ is unknown and must be replaced by a plug-in estimator $\hat{\mathbf{B}}$. This substitution introduces two distinct types of estimation error: a simple perturbation $\boldsymbol{\Delta}$, arising when $\hat{\mathbf{B}}$ replaces $\mathbf{B}$ only in the numerator, and a composite perturbation $\tilde{\boldsymbol{\Delta}}$, arising when the replacement occurs in both the numerator and the denominator. Under both perturbation regimes, we decompose the total sum of squares into three components and conduct a detailed analysis of their asymptotic properties. This reveals a key, and perhaps surprising, distinction between simple and composite perturbations: the cross term $\tr({\A^*}\bDelta)$ is asymptotically negligible, whereas its composite counterpart $\tr({\A^*}\tilde{\bDelta})$ is not. Motivated by this, we develop a unified decomposition framework, expressing the composite perturbation matrix as $\tilde{\bDelta}=\check{\A}+\bDelta+\check{\bDelta}$, where $\check{\A}$ is a bias matrix of the normalized adjacency matrix, $\bDelta$ is the simple perturbation, and $\check{\bDelta}$ is a bias matrix of $\bDelta$. This structured decomposition allows us to precisely isolate and control each source of error, leading to a refined limiting theory for two key classes of test statistics. Concretely, for the largest eigenvalue statistic, we improve the existing condition from $K=O(n^{1/6-\tau})$ to the optimal rate $K=o(n^{1/6})$ under both simple and composite perturbations. For the linear spectral statistic, our unified decomposition framework provides the necessary structure to systematically control these errors term by term, leading to a complete and rigorous proof of asymptotic normality.

Deployed prediction systems are often retrained on fixed calendars, even when model staleness and retraining burden vary over time. This short communication formulates retraining for Bayesian prediction systems as a cost-sensitive predictive-regret decision. The central monitoring state is posterior learning debt, defined as the Kullback–Leibler divergence from a reference shadow posterior to the deployed frozen posterior. In the decision layer, a retraining cost is compared with the expected one-period predictive regret of waiting. A continuous-severity version retrains when calibrated expected regret exceeds the retraining cost, while the familiar two-state excess-loss rule is a special case. The empirical study is an exact-state proof-of-concept in a synthetic conjugate simulation with warm-started deployed and shadow normal-inverse-gamma posteriors, separate update, monitoring, and evaluation batches, lagged deployment actions, expanded baseline grids, and score-unit sensitivity. Under the primary 75th-percentile score-unit scaling, an age-adjusted debt-threshold policy improves on tuned calendar retraining in all 72 non-stable scenario cells and on tuned CUSUM in 58 of 72 cells, with mean relative objectives 0.677 and 0.975, respectively. Debt-utility and hybrid-utility policies also improve strongly over tuned calendar retraining, but they do not dominate tuned CUSUM. Median and mean score-unit sensitivities show the same main calendar result, while the CUSUM comparison remains policy-dependent. The contribution is a transparent decision layer for deployed Bayesian prediction systems, not a universal replacement for drift detection.

This paper proposes niching importance sampling, a framework that combines concepts from reliability analysis, e.g. Markov chains, importance sampling, and relative cross entropy minimisation, with niching techniques from evolutionary multi-modal optimisation. The result is a highly robust estimator of the probability of failure, that can tackle sampling challenges posed by the underlying geometry of a reliability problem. Niching importance sampling is tested on a range of numerical examples and is shown to consistently avoid the degenerate behaviour observed for existing reliability methods on several multi-modal performance functions.

Causal mediation analysis has become an important and increasingly used framework for evaluating candidate immune response biomarkers in vaccine research. A controlled effects approach has been proposed to estimate controlled risk curves under a counterfactual scenario in which the entire study population is vaccinated and their post-vaccination immune responses are set to a range of fixed levels. This framework performs well when the study population is antigenically naïve, that is, individuals have not been previously exposed to the antigen, as is common in HIV-1 vaccine research and during the early phases of the COVID-19 pandemic. However, the controlled effects framework becomes more challenging to apply in antigen-experienced populations, where prior vaccination or infection has occurred, as in the case of influenza, dengue, and more recent phases of the COVID-19 pandemic. In such settings, a key identification assumption for valid causal mediation analysis, the positivity assumption, is violated: it is no longer plausible to conceive of a hypothetical intervention that sets a post-vaccination immune marker to a fixed level below an individual's baseline immune level. In this article, we introduce a weighted controlled risk approach that targets a subpopulation for whom there is a prespecified probability of attaining a post-vaccination immune marker level. We further generalize this framework to study contrasts of controlled risks for relevant subpopulations. We demonstrate the validity of the proposed estimators through simulation studies and apply the method to reanalyze post-vaccination neutralizing antibody titers against Omicron BA.4/BA.5 as an immune correlate of COVID-19 in the Coronavirus Variant Immunologic Landscape (COVAIL) trial. R code to implement the proposed method can be found on Github: https://github.com/Qijia-He/weighted_CVE.

Spiking reservoir computing provides an energy-efficient approach to temporal processing, but reliably tuning reservoirs to operate at the edge-of-chaos is challenging due to experimental uncertainty. This work bridges abstract notions of criticality and practical stability by introducing and exploiting the robustness interval, an operational measure of the hyperparameter range over which a reservoir maintains performance above task-dependent thresholds. Through systematic evaluations of Leaky Integrate-and-Fire (LIF) architectures on both static (MNIST) and temporal (synthetic Ball Trajectories) tasks, we identify consistent monotonic trends in the robustness interval across a broad spectrum of network configurations: the robustness-interval width decreases with presynaptic connection density $\beta$ (i.e., directly with sparsity) and directly with the firing threshold $\theta$. We further identify specific $(\beta, \theta)$ pairs that preserve the analytical mean-field critical point $w_{\text{crit}}$, revealing iso-performance manifolds in the hyperparameter space. Control experiments on Erdős-Rényi graphs show the phenomena persist beyond small-world topologies. Finally, our results show that $w_{\text{crit}}$ consistently falls within empirical high-performance regions, validating $w_{\text{crit}}$ as a robust starting coordinate for parameter search and fine-tuning. To ensure reproducibility, the full Python code is publicly available.

Classical multivariate shape analysis relies on covariance-standardized moments, such as Mardia skewness and kurtosis, which are sensitive to outliers and require finite moments. This paper introduces vector median absolute deviation (VMedAD) moments for robust multivariate shape analysis. The proposed framework replaces moment aggregation and covariance standardization with median-based center-outward contrasts defined through data depth, yielding affine equivariance and moment-free vector moments. VMedAD moments provide direction-preserving measures of multivariate skewness and directional peripheral dominance, separating central structure from tail-driven behavior. Consistency, breakdown properties, and affine equivariance are established, and simulation and real dataset examples demonstrate improved robustness and geometric interpretability over classical and projection-based methods.

Deep linear networks (DLNs) are used as an analytically tractable model of the training dynamics of deep neural networks. While gradient descent in DLNs is known to exhibit saddle-to-saddle dynamics, the impact of stochastic gradient descent (SGD) noise on this regime remains poorly understood. We investigate the dynamics of SGD during training of DLNs in the saddle-to-saddle regime. We model the training dynamics as stochastic Langevin dynamics with anisotropic, state-dependent noise. Under the assumption of aligned and balanced weights, we derive an exact decomposition of the dynamics into a system of one-dimensional per-mode stochastic differential equations. This establishes that the maximal diffusion along a mode precedes the corresponding feature being completely learned. We also derive the stationary distribution of SGD for each mode: in the absence of label noise, its marginal distribution along specific features coincides with the stationary distribution of gradient flow, while in the presence of label noise it approximates a Boltzmann distribution. Finally, we confirm experimentally that the theoretical results hold qualitatively even without aligned or balanced weights. These results establish that SGD noise encodes information about the progression of feature learning but does not fundamentally alter the saddle-to-saddle dynamics.

We study mean estimation of a random vector $X$ in a distributed parameter-server-worker setup. Worker $i$ observes samples of $a_i^\top X$, where $a_i^\top$ is the $i$th row of a known sensing matrix $A$. The key challenges are adversarial measurements and asynchrony: a fixed subset of workers may transmit corrupted measurements, and workers are activated asynchronously–only one is active at any time. In our previous work, we proposed a two-timescale $\ell_1$-minimization algorithm and established asymptotic recovery under a null-space-property-like condition on $A$. In this work, we establish tight non-asymptotic convergence rates under the same null-space-property-like condition. We also identify relaxed conditions on $A$ under which exact recovery may fail but recovery of a projected component of $\mathbb{E}[X]$ remains possible. Overall, our results provide a unified finite-time characterization of robustness, identifiability, and statistical efficiency in distributed linear estimation with adversarial workers, with implications for network tomography and related distributed sensing problems.

We derive generalization error bounds for the training of two-layer neural networks without assuming boundedness of the loss function, using Wasserstein distance estimates on the discrepancy between a probability distribution and its associated empirical measure, together with moment bounds for the associated stochastic gradient method. In the case of independent test data, we obtain a dimension-free rate of order $O(n^{-1/2} )$ on the $n$-sample generalization error, whereas without independence assumption, we derive a bound of order $O(n^{-1 / ( d_{\rm in}+d_{\rm out} )} )$, where $d_{\rm in}$, $d_{\rm out}$ denote input and output dimensions. Our bounds and their coefficients can be explicitly computed prior to the training of the model, and are confirmed by numerical simulations.

Small regional datasets pose a dual statistical problem: correlated predictors inflate estimation variance, while flexible learners can become unstable because the available information per adaptive degree of freedom is limited. We examine this issue through predictive volatility, defined as the cross-sample dispersion and upper-tail behaviour of out-of-sample loss. Using simulation evidence reported for sparse linear, near-linear and heavy-tailed settings, we compare ordinary least squares, frequentist penalties, Bayesian shrinkage models, bounded-response and spatial specifications, and flexible machine-learning procedures. In the reported simulation results, regularised linear estimators generally dominate in the linear high-collinearity micro-sample settings and remain the most reliable overall, whereas tree-based methods become more competitive only when the signal is weakly nonlinear and the sample size is larger. In the empirical application to 34 Indonesian provinces, ridge yields the best leave-one-out performance, followed by elastic net and lasso. Across the Bayesian shrinkage specifications, ICT skills show the most consistent negative association with poverty, with the strongest support under horseshoe and spike-and-slab formulations. These results suggest that, in micro-sample regional modelling, the main constraint is limited information per effective degree of freedom rather than insufficient algorithmic flexibility.

The static “train then deploy" paradigm fundamentally limits Large Language Models (LLMs) from dynamically adapting their weights in response to continuous streams of new information inherent in real-world tasks. Test-Time Training (TTT) offers a compelling alternative by updating a subset of model parameters (fast weights) at inference time, yet its potential in the current LLM ecosystem is hindered by critical barriers including architectural incompatibility, computational inefficiency and misaligned fast weight objectives for language modeling. In this work, we introduce In-Place Test-Time Training (In-Place TTT), a framework that seamlessly endows LLMs with Test-Time Training ability. In-Place TTT treats the final projection matrix of the ubiquitous MLP blocks as its adaptable fast weights, enabling a “drop-in" enhancement for LLMs without costly retraining from scratch. Furthermore, we replace TTT's generic reconstruction objective with a tailored, theoretically-grounded objective explicitly aligned with the Next-Token-Prediction task governing autoregressive language modeling. This principled objective, combined with an efficient chunk-wise update mechanism, results in a highly scalable algorithm compatible with context parallelism. Extensive experiments validate our framework's effectiveness: as an in-place enhancement, it enables a 4B-parameter model to achieve superior performance on tasks with contexts up to 128k, and when pretrained from scratch, it consistently outperforms competitive TTT-related approaches. Ablation study results further provide deeper insights on our design choices. Collectively, our results establish In-Place TTT as a promising step towards a paradigm of continual learning in LLMs.

In a fully-Bayesian Functional Principal Components Analysis (FPCA) the principal components are treated as unknown infinite-dimensional parameters. By projecting the functional principal components on a rich orthonormal spline basis, we show that orthonormality of the principal components is equivalent to orthonormality of the spline coefficients. A penalty on the integral of the second derivative of the functional principal components can be induced on the spline coefficients, where each function has its own smoothing parameter. Finally, each smoothing parameter is treated as an inverse variance component in the associated mixed effects model. In this work, we demonstrate that no additional conditions are required to ensure that the corresponding smoothing prior, and thus the posterior distribution, is proper. This allows the choice of less informative priors, such that smoothing is driven by the data.

Estimating Conditional Average Treatment Effects (CATE) at the individual level is central to precision marketing, yet systematic benchmarking of uplift modeling methods at industrial scale remains limited. We present UpliftBench, an empirical evaluation of four CATE estimators: S-Learner, T-Learner, X-Learner (all with LightGBM base learners), and Causal Forest (EconML), applied to the Criteo Uplift v2.1 dataset comprising 13.98 million customer records. The near-random treatment assignment (propensity AUC = 0.509) provides strong internal validity for causal estimation. Evaluated via Qini coefficient and cumulative gain curves, the S-Learner achieves the highest Qini score of 0.376, with the top 20% of customers ranked by predicted CATE capturing 77.7% of all incremental conversions, a 3.9x improvement over random targeting. SHAP analysis identifies f8 as the dominant heterogeneous treatment effect (HTE) driver among the 12 anonymized covariates. Causal Forest uncertainty quantification reveals that 1.9% of customers are confident persuadables (lower 95% CI > 0) and 0.1% are confident sleeping dogs (upper 95% CI < 0). Our results provide practitioners with evidence-based guidance on method selection for large-scale uplift modeling pipelines.

Financial statement auditors use a risk-based approach to evidence collection to obtain reasonable assurance. When an initial sample does not support a conclusion, they may inspect additional items. We develop a sampling procedure by formulating this additional inspection as sequential hypothesis testing for a fixed finite population sampled without replacement. A tolerable deviation rate and a margin below that rate define the composite hypotheses and the intervening indifference region, and lower and upper stopping boundaries control the two probabilities of incorrect conclusions. Under simple random sampling, a hypergeometric recursion computes the boundaries, decision probabilities, and expected sample size exactly. The procedure permits decisions at prespecified times, including fixed intervals. A separate extension imposes a fixed maximum sample size. We also consider one-sided error control and complete examination of selected items. When the state is multidimensional or has too many possible values for direct enumeration, Monte Carlo simulation constructs candidate boundaries. An independent simulation then provides one-sided upper confidence bounds for the two error probabilities at the least-favorable population counts.

Representation of cities as organisms with metabolic processes is a useful analogy for urban design, development and sustainability. Urban metabolism can be modeled by representing urban systems as networks. The various networks included in a city's metabolism are interdependent in complex ways. Thus, understanding the interaction among these networks is essential to understanding how a healthy urban metabolism is sustained and how injuries to the metabolic system can "heal". It is particularly important to understand how disruptions to one system in an urban area affect the functioning of other systems. Using distribution-level data from a real U.S. city on the electricity distribution system and road geometry, we apply connected network modeling to two critical inter-connected urban infrastructure sectors: energy and transportation. We quantify the robustness of these interdependent networks by evaluating the connectivity disruptions that may occur due to natural or synthetic disruptive events, using both unweighted and weighted metrics.

Under general assumptions on the target distribution $p^\star$, we establish a sharp Lipschitz regularity theory for flow-matching vector fields and diffusion-model scores, with optimal dependence on time and dimension. As applications, we obtain Wasserstein discretization bounds for Euler-type samplers in dimension $d$: with $N$ discretization steps, the error achieves the optimal rate $\sqrt{d}/N$ up to logarithmic factors. Moreover, the constants do not deteriorate exponentially with the spatial extent of $p^\star$. We also show that the one-sided Lipschitz control yields a globally Lipschitz transport map from the standard Gaussian to $p^\star$, which implies Poincaré and log-Sobolev inequalities for a broad class of probability measures.

In Bayesian inference for the Cox proportional hazards model, modeling the baseline hazard function is challenging. Recently, direct Bayesian inference using the partial likelihood is considered in the framework of general Bayesian inference. In terms of posterior computation, several studies have examined sampling algorithms under the Cox model. In this study, we propose two Gibbs sampling algorithms for Bayesian inference in the Cox proportional hazards model, motivated by a rank-ordered data representation and based on the Plackett–Luce and generalized Plackett–Luce models with P'{o}lya–Gamma data augmentation, referred to as PL-Cox and GPL-Cox, respectively. The two proposed methods offer practical advantages, as they do not require correction of posterior samples, naturally handle tied event times, and are readily extensible to shared frailty models. In simulation study, we considered multiple survival model settings, including continuous and discrete survival time models, as well as scenarios with varying degrees of ties, and found that the PL-Cox model exhibited relatively stable performance. In analyses of a large real dataset, the proposed methods remained computationally feasible, and the GPL-Cox model showed more favorable computational scalability than the PL-Cox model. In analyses of real data incorporating shared frailty, both methods demonstrated good computational efficiency.

Neural network classifiers trained with cross-entropy loss achieve strong predictive accuracy but lack the capability to provide inherent predictive uncertainty estimates, thus requiring external techniques to obtain these estimates. In addition, softmax scores for the true class can vary substantially across independent training runs, which limits the reliability of uncertainty-based decisions in downstream tasks. Evidential Deep Learning aims to address these limitations by producing uncertainty estimates in a single pass, but evidential training is highly sensitive to design choices including loss formulation, prior regularization, and activation functions. Therefore, this work introduces an alternative Dirichlet parameter estimation strategy by applying a method of moments estimator to ensembles of softmax outputs, with an optional maximum-likelihood refinement step. This ensemble-based construction decouples uncertainty estimation from the fragile evidential loss design while also mitigating the variability of single-run cross-entropy training, producing explicit Dirichlet predictive distributions. Across multiple datasets, we show that the improved stability and predictive uncertainty behavior of these ensemble-derived Dirichlet estimates translate into stronger performance in downstream uncertainty-guided applications such as prediction confidence scoring and selective classification.

We investigate the data distribution valuation problem, which aims to quantify the values of data distributions from their samples. This is a recently proposed problem that is related to but different from classical data valuation and can be applied to various applications. For this problem, we develop a novel framework called Generalized Bayes Valuation that utilizes generalized Bayesian inference with a loss constructed from transferability measures. This framework allows us to solve, in a unified way, seemingly unrelated practical problems, such as annotator evaluation and data augmentation. Using the Bayesian principles, we further improve and enhance the applicability of our framework by extending it to the continuous data stream setting. Our experiment results confirm the effectiveness and efficiency of our framework in different real-world scenarios.

In ecological studies niche overlap is often used to quantify species interaction and dynamics. This paper develops a robust, nonparametric statistical framework for quantifying and analyzing multivariate niche overlap. Parametric methods are often constrained by restrictive assumptions and tend to underperform in complex multivariate settings. We introduce a nonparametric overlap index and propose estimators for it. Further, we investigate asymptotic properties of the estimators. We also propose bootstrap-based inference procedures that enable statistical testing and simultaneous confidence intervals in small sample settings. Extensive numerical examples demonstrate that our proposed methods maintain correct size and exhibit robust power across various scenarios. We illustrate the practical utility of our methodology using stable isotope measurements from multiple fish species and provide distinct ecological insights regarding species niche differentiation.

We study a two-dimensional incompressible vorticity equation on the torus driven by transport-type fractional Brownian noise with Hurst parameter $H \in (1/2,1)$. The model captures persistent, long-range correlated forcing consistent with inertial-range scaling laws and fractional Brownian approximations of turbulent fluctuations. A central ingredient of our approach is a version of the sewing lemma adapted to a class of integrands that includes, but is not limited to, transport-type structures. This result provides a flexible tool for constructing the Young integral and serves as a basis for analysing a wider class of stochastic partial differential equations. Using this approach, we establish existence and uniqueness of solutions via a fixed point argument and investigate statistical properties of the flow. In particular, we study quadratic functionals of the solution and derive an estimator for the Hurst parameter $H$.

Mixture of linear regression is well studied in statistics and machine learning, where the data points are generated probabilistically using $k$ linear models. Algorithms like Expectation Maximization (EM) may be used to recover the ground truth regressors for this problem. Recently, in \cite{pal2022learning,ghosh_agnostic} the mixed linear regression problem is studied in the agnostic setting, where no generative model on data is assumed. Rather, given a set of data points, the objective is fit $k$ lines by minimizing a suitable loss function. It is shown that a modification of EM, namely gradient EM converges exponentially to appropriately defined loss minimizer even in the agnostic setting. In this paper, we study the problem of fitting $k$ parametric functions to given set of data points. We adhere to the agnostic setup. However, instead of fitting lines equipped with quadratic loss, we consider any arbitrary parametric function fitting equipped with a strongly convex and smooth loss. This framework encompasses a large class of problems including mixed linear regression (regularized), mixed linear classifiers (mixed logistic regression, mixed Support Vector Machines) and mixed generalized linear regression. We propose and analyze gradient EM for this problem and show that with proper initialization and separation condition, the iterates of gradient EM converge exponentially to appropriately defined population loss minimizers with high probability. This shows the effectiveness of EM type algorithm which converges to optimal solution in the non-generative setup beyond mixture of linear regression.

Count-weighted temporal networks often exhibit unequal dispersion in the edge weights, which cannot be fully explained by modelling observational heterogeneity through latent factors in the conditional mean. Therefore, we propose new dynamic network model classes exploiting the Generalized Poisson distribution to capture both under- and overdispersion. We consider three different dynamic specifications: latent factor dynamics, autoregressive dynamics, and latent position dynamics, and study some theoretical properties of the random networks, showing the impact of the dispersion parameter on the random network's connectivity. After discussing the parameter identification strategy, we present a Bayesian inference procedure along with a posterior sampling algorithm. A numerical illustration demonstrates the effectiveness of the designed algorithm and provides estimates of the misspecification bias when unequal dispersion is neglected. Our new models are then applied to two relevant dynamic datasets considered in previous studies: a set of bike-sharing dynamic networks and a set of dynamic media networks. Our results highlight the importance of explicitly modeling overdispersion for both an accurate in-sample fit and out-of-sample performance.

Approaches to bivariate causal discovery based on the minimum description length (MDL) principle approximate the (uncomputable) Kolmogorov complexity of the models in each causal direction, selecting the one with the lower total complexity. The premise is that nature's mechanisms are simpler in their true causal order. Inherently, the description length (complexity) in each direction includes the description of the cause variable and that of the causal mechanism. In this work, we argue that current state-of-the-art MDL-based methods do not correctly address the problem of estimating the description length of the cause variable, effectively leaving the decision to the description length of the causal mechanism. Based on rate-distortion theory, we propose a new way to measure the description length of the cause, corresponding to the minimum rate required to achieve a distortion level representative of the underlying distribution. This distortion level is deduced using rules from histogram-based density estimation, while the rate is computed using the related concept of information dimension, based on an asymptotic approximation. Combining it with a traditional approach for the causal mechanism, we introduce a new bivariate causal discovery method, termed rate-distortion MDL (RDMDL). We show experimentally that RDMDL achieves competitive performance on the Tübingen dataset. All the code and experiments are publicly available at github.com/tiagobrogueira/Causal-Discovery-In-Exchangeable-Data.