Papers

Diffusion models provide an expressive framework for sampling from complex, unnormalized distributions. In this work, we extend Maximum Entropy Reinforcement Learning (ME-RL) to diffusion-based policies by introducing Diffusion-Augmented Markov Decision Processes (DA-MDPs). DA-MDPs interpret each reverse-diffusion transition as an individual reinforcement-learning decision, while only the final denoised action is executed in the environment. Our DA-MDPs follow from a principled derivation based on the variational-inference formulation of ME-RL. By augmenting policy and target trajectories with intermediate diffusion variables, we obtain a tractable reverse-KL upper bound via the data-processing inequality. This bound decomposes across denoising transitions, yielding diffusion-augmented variants of soft rewards, value functions, and local policy objectives. This provides a general framework for adapting ME-RL algorithms to diffusion policies while differentiating through only one diffusion transition at a time. We instantiate the framework with PPO, REPPO, and a maximum-entropy extension of WPO. Experiments demonstrate improved continuous-control performance, benefits from additional diffusion steps, and memory-efficient training. On the StackCube and PushT manipulation tasks, DA-MDP methods learn alternative successful strategies from the same initial state and achieve higher success rates and generally higher success-weighted mode entropy than the Gaussian ME-RL baseline. We also demonstrate successful training when using action chunking.

Characteristics of the West African Monsoon (WAM) season, such as its onset and dry spell occurrences, are notoriously difficult to predict. However, these characteristics are key indicators farmers use to decide when to plant crops, having a major influence on their overall yield. While many studies have shown correlations between global sea surface temperatures and characteristics of the WAM season, there are few that effectively implement this information into machine learning (ML) prediction models. This study is focused on predicting dry spells, that is, if there will be a period of consecutive days without rain after the onset of the WAM. We first investigated the best ways to define onset and dry spells and gathered sea surface temperature training data from both real-world observations and a climate simulation model. Then we constructed an adaptive-threshold logistic regression model for dry spell prediction, to which we applied a custom feature selection method and spatial regularization. Using Leave-One-Out cross validation testing, we found significant results in multiple binary classification metrics. These models overcome some limitations that current approaches have, such as being computationally intensive and needing bias correction. We also aim for this study to serve as a framework for ML use in the context of targeted prediction of certain weather phenomena using climatologically relevant variables.

In this paper, we explore how different selections of basis functions impact the efficacy of frequency domain techniques in statistical independence tests, and study different algorithms for extracting low-dimensional algebraic relations from dependent data. We examine a range of complete orthonormal bases functions including the Legendre polynomials, Fourier series, Walsh functions, and standard and nonstandard Haar wavelet bases. We utilize fast transformation algorithms to efficiently transform physical domain data to frequency domain coefficients. The main focuses of this paper are the effectiveness of different basis selections in detecting data dependency using frequency domain data, e.g., whether varying basis choices significantly influence statistical power loss for small data with large noise; and on the stability of different optimization formulations for finding proper algebraic relations when data are dependent. We present numerical results to demonstrate the effectiveness of frequency domain-based statistical analysis methods and provide guidance for selecting the proper basis and algorithm to detect a particular type of relations.

Data collected by the Interstellar Boundary Explorer (IBEX) satellite, recording heliospheric energetic neutral atoms (ENAs), exhibit a phenomenon that has caused space scientists to revise hypotheses about the physical processes, and computer simulations under those models, in play at the boundary of our solar system. Evaluating the fit of these computer models involves tuning their parameters to observational data from IBEX. This would be a classic (Bayesian) inverse problem if not for three challenges: (1) the computer simulations are slow, limiting the size of campaigns of runs; so (2) surrogate modeling is essential, but outputs are high-resolution images, thwarting conventional methods; and (3) IBEX observations are counts, whereas most inverse problem techniques assume Gaussian field data. To fill that gap we propose a novel approach to Bayesian inverse problems coupling a Poisson response with a sparse Gaussian process surrogate using the Vecchia approximation. We demonstrate the capabilities of our proposed framework, which compare favorably to alternatives, through multiple simulated examples in terms of recovering "true" computer model parameters and accurate out-of-sample prediction. We then apply this new technology to IBEX satellite data and associated computer models developed at Los Alamos National Laboratory.

Safety-critical environments are inherently dynamic. Distribution shifts, emerging vulnerabilities, and evolving requirements demand continuous updates to machine learning models. Yet even benign parameter updates can have unintended consequences, such as catastrophic forgetting in classical models or alignment drift in foundation models. Existing heuristic approaches (e.g., regularization, parameter isolation) can mitigate these effects but cannot certify that updated models continue to satisfy required performance specifications. We address this problem by introducing a framework for provably safe model updates. Our approach first formalizes the problem as computing the largest locally invariant domain (LID): a connected region in parameter space where all points are certified to satisfy a given specification. While exact maximal LID computation is intractable, we show that relaxing the problem to parameterized abstract domains (orthotopes, zonotopes) yields a tractable primal-dual formulation. This enables efficient certification of updates - independent of the data or algorithm used - by projecting them onto the safe domain. Our formulation further allows computation of multiple approximately optimal LIDs, incorporation of regularization-inspired biases, and use of lookahead data buffers. Across continual learning and foundation model fine-tuning benchmarks, our method matches or exceeds heuristic baselines for avoiding forgetting while providing formal safety guarantees.

A central challenge in machine learning is to distinguish genuine structure from chance correlations in high-dimensional data. In this work, we address this issue for the perceptron, a foundational model of neural computation. Specifically, we investigate the relationship between the pattern load $\alpha$ and the variable selection ratio $\rho$ for which a simple perceptron can perfectly classify $P = \alpha N$ random patterns by optimally selecting $M = \rho N$ variables out of $N$ variables. While the Cover–Gardner theory establishes that a random subset of $\rho N$ dimensions can separate $\alpha N$ random patterns if and only if $\alpha < 2\rho$, we demonstrate that optimal variable selection can surpass this bound by developing a method, based on the replica method from statistical mechanics, for enumerating the combinations of variables that enable perfect pattern classification. This not only provides a quantitative criterion for distinguishing true structure in the data from spurious regularities, but also yields the storage capacity of associative memory models with sparse asymmetric couplings.

Finite mixture models are frequently used to uncover latent structures in high-dimensional datasets (e.g.\ identifying clusters of patients in electronic health records). The inference of such structures can be performed in a Bayesian framework, and involves the use of sampling algorithms such as Gibbs samplers aimed at deriving posterior distribution of the probabilities of observations to belong to specific clusters. Unfortunately, traditional implementations of Gibbs samplers in this context often face critical challenges, such as inefficient use of computational resources and unnecessary updates for observations that are highly likely to remain in their current cluster. This paper introduces a new adaptive Gibbs sampler that improves the convergence efficiency over existing methods. In particular, our sampler is guided by a function that, at each iteration, uses the past of the chain to focus the updating on observations potentially misclassified in the current clustering, i.e.\ those with a low probability of belonging to their current component. Through simulation studies and two real data analyses, we empirically demonstrate that, in terms of convergence time, our method tends to perform more efficiently compared to state-of-the-art approaches.

Given observations from a positive random variable contaminated by multiplicative measurement error, we consider a nonparametric goodness-of-fit testing task for its unknown density in a non-asymptotic framework. We propose a testing procedure based on estimating a quadratic functional of the Mellin transform of the unknown density and the null. We derive non-asymptotic testing radii and testing rates over Mellin-Sobolev spaces, which naturally characterize regularity and ill-posedness in this model. By employing a multiple testing procedure with Bonferroni correction, we obtain data-driven procedures and analyze their performance. Compared with the non-adaptive tests, their testing radii deteriorate by at most a logarithmic factor. We illustrate the testing procedures with a simulation study using various choices of densities.

The K function and its related statistics have been an enduring tool in the analysis of spatial point processes, providing an easy to compute and interpret summary statistic for characterising the interactions between points of one type, or between two different types of points. In this paper, we introduce a partial K function, enabling us to account for some of the effects of the other point types when analysing point-point interactions. The partial K function we introduce reduces to the usual K function when the other points are independent of the points of interest and has a similar interpretation. Using examples, we demonstrate how the partial K function can unpick dependence between point types that would otherwise be hidden in the usual K function. We also discuss important bias correction steps and hyperparameter selection. In addition, we introduce an extension to account for other spatial covariates, and demonstrate the methodology on the Lansing Woods dataset.

Diffusion generative models have emerged as powerful tools for producing synthetic data from an empirically observed distribution. A common approach involves simulating the time-reversal of an Ornstein-Uhlenbeck (OU) process initialized at the true data distribution. Since the score function associated with the OU process is typically unknown, it is approximated using a trained neural network. This approximation, along with finite time simulation, time discretization and statistical approximation, introduce several sources of error whose impact on the generated samples must be carefully understood. Previous analyses have quantified the error between the generated and the true data distributions in terms of Wasserstein distance or Kullback-Leibler (KL) divergence. However, both metrics present limitations: KL divergence requires absolute continuity between distributions, while Wasserstein distance, though more general, leads to error bounds that scale poorly with dimension, rendering them impractical in high-dimensional settings. In this work, we derive an explicit, dimension-free bound on the discrepancy between the generated and the true data distributions. The bound is expressed in terms of a smooth test functional with bounded first and second derivatives. The key novelty lies in the use of this weaker, functional metric to obtain dimension-independent guarantees, at the cost of higher regularity on the test functions. As an application, we formulate and solve a variational problem to minimize the time-discretization error, leading to the derivation of an optimal time-scheduling strategy for the reverse-time diffusion. Interestingly, this scheduler has appeared previously in the literature in a different context; our analysis provides a new justification for its optimality, now grounded in minimizing the discretization bias in generative sampling.

Decision trees and random forest remain highly competitive for classification on medium-sized, standard datasets due to their robustness, minimal preprocessing requirements, and interpretability. However, a single tree suffers from high estimation variance, while large ensembles reduce this variance at the cost of substantial computational overhead and diminished interpretability. In this paper, we propose Decision Tree Embedding (DTE), a fast and effective method that leverages the leaf partitions of a trained classification tree to construct an interpretable feature representation. By using the sample means within each leaf region as anchor points, DTE maps inputs into an embedding space defined by the tree's partition structure, effectively circumventing the high variance inherent in decision-tree splitting rules. We further introduce an ensemble extension based on additional bootstrap trees, and pair the resulting embedding with linear discriminant analysis for classification. We establish several population-level theoretical properties of DTE, including its preservation of conditional density under mild conditions and a characterization of the resulting classification error. Empirical studies on synthetic and real datasets demonstrate that DTE strikes a strong balance between accuracy and computational efficiency, outperforming or matching random forest and shallow neural networks while requiring only a fraction of their training time in most cases. Overall, the proposed DTE method can be viewed either as a scalable decision tree classifier that improves upon standard split rules, or as a neural network model whose weights are learned from tree-derived anchor points, achieving an intriguing integration of both paradigms.

(This is the third version of a working paper.) We develop a family of self-normalized concentration inequalities for marginal mean under martingale-difference structure and $\phi/\tilde{\phi}$-mixing conditions, where the latter includes many processes that are not strongly mixing. The variance term is fully data-observable: naive sample variance in the martingale case and an empirical block long-run variance under mixing conditions. Thus, no predictable variance proxy is required. No specific assumption on the decay of the mixing coefficients (e.g. summability) is needed for the validity. The constants are explicit and the bounds are ready to use.

In this paper, we investigate a second-order stochastic algorithm for solving large-scale binary classification problems. We propose to make use of a new hybrid stochastic Newton algorithm that includes two weighted components in the Hessian matrix estimation: the first one coming from the natural Hessian estimate and the second associated with the stochastic gradient information. Our motivation comes from the fact that both parts evaluated at the true parameter of logistic regression, are equal to the Hessian matrix. This new formulation has several advantages and it enables us to prove the almost sure convergence of our stochastic algorithm to the true parameter. Moreover, we significantly improve the almost sure rate of convergence to the Hessian matrix. Furthermore, we establish the central limit theorem for our hybrid stochastic Newton algorithm. Finally, we show a surprising result on the almost sure convergence of the cumulative excess risk.

Approximating integrals is a fundamental task in probability theory and statistical inference, and their applied fields of signal processing, and Bayesian learning, as soon as expectations over probability distributions must be computed efficiently and accurately. When these integrals lack closedform expressions, numerical methods must be used, from the Newton-Cotes formulas and Gaussian quadrature, to Monte Carlo and variational approximation techniques. Despite these numerous tools, few are guaranteed to preserve majoration/minoration inequalities, while this feature is fundamental in certain applications in statistics. In this paper, we focus on the integration problem arising in the estimation of moments of scalar unnormalized distributions. We introduce a sequential method for constructing upper and lower bounds on the sought integral. Our approach leverages the majorization-minimization framework to iteratively refine these bounds using an envelope principle. The method has proven convergence and controlled accuracy under mild conditions. We then generalize the method to the multi-dimensional setting, along with an effective implementation strategy based on power diagrams. We demonstrate the effectiveness of the proposed approach through a detailed numerical example of the estimation of a Monte Carlo sampler variance in a Bayesian inference problem, in one- and two-dimensional cases.

Bipartite networks are widely used to encode the ecological interactions. Being able to compare the organization of bipartite networks is a first step toward a better understanding of how environmental factors shape community structure and resilience. Yet current methods for structure detection in bipartite networks overlook shared patterns across collections of networks. We introduce the colBiSBM, a family of probabilistic models for collections of bipartite networks that extends the classical Latent Block Model (LBM). The proposed framework assumes that networks are independent realizations of a shared mesoscale structure, encoded through common inter-block connectivity parameters. We establish identifiability conditions for the different variants of colBiSBM and develop a variational EM algorithm for parameter estimation, coupled with an adaptation of the Integrated Classification Likelihood (ICL) criterion for model selection. We demonstrate how our approach can be used to classify networks based on their topology or organization. Simulation studies highlight the ability of colBiSBM to recover common structures, improve clustering performance, and enhance link prediction by borrowing strength across networks. An application to plant–pollinator networks highlights how the method uncovers shared ecological roles and partitions networks into sub-collections with similar connectivity patterns. These results illustrate the methodological and practical advantages of joint modeling over separate network analyses in the study of bipartite systems.

Learning the structure of a Bayesian network from decentralized data poses two major challenges: (i) ensuring rigorous privacy guarantees for participants, and (ii) avoiding communication costs that scale poorly with dimensionality. In this work, we introduce Fed-Sparse-BNSL, a novel federated method for learning linear Gaussian Bayesian network structures that addresses both challenges. By combining differential privacy with greedy updates that target only a few relevant edges per participant, Fed-Sparse-BNSL efficiently uses the privacy budget while keeping communication costs low. Our careful algorithmic design preserves model identifiability and enables accurate structure estimation. Experiments on synthetic and real datasets demonstrate that Fed-Sparse-BNSL achieves utility close to non-private baselines while offering substantially stronger privacy and communication efficiency.

L. Klebanov proved the following theorem. Let $\xi_1, \dots, \xi_n$ be independent random variables. Consider linear forms $L_1=a_1\xi_1+\cdots+a_n\xi_n,$ $L_2=b_1\xi_1+\cdots+b_n\xi_n,$ $L_3=c_1\xi_1+\cdots+c_n\xi_n,$ $L_4=d_1\xi_1+\cdots+d_n\xi_n,$ where the coefficients $a_j, b_j, c_j, d_j$ are real numbers. If the random vectors $(L_1,L_2)$ and $(L_3,L_4)$ are identically distributed, then all $\xi_i$ for which $a_id_j-b_ic_j\neq 0$ for all $j=\overline{1,n}$ are Gaussian random variables. The present article is devoted to an analogue of the Klebanov theorem in the case when random variables take values in the group $\mathbb{R}\times \mathbb{Z}(2)$ and the coefficients of the linear forms are topological endomorphisms of this group.

Bayesian inference is optimal when the statistical model is well-specified, while outside this setting Bayesian inference can catastrophically fail; accordingly a wealth of post-Bayesian methodologies have been proposed. Predictively oriented (PrO) approaches lift the statistical model $P_\theta$ to an (infinite) mixture model $\int P_\theta \; \mathrm{d}Q(\theta)$ and fit this predictive distribution via minimising an entropy-regularised objective functional. In the well-specified setting one expects the mixing distribution $Q$ to concentrate around the true data-generating parameter in the large data limit, while such singular concentration will typically not be observed if the model is misspecified. Our contribution is to demonstrate that one can empirically detect model misspecification by comparing the standard Bayesian posterior to the PrO `posterior' $Q$, and that each can be efficiently computed via variational gradient descent. A simulation study confirms that model misspecification can be rigorously tested using this framework, and a detailed case study in seismic tomography illustrates the potential of this approach as a novel and widely-applicable diagnostic tool for the standard Bayesian workflow.

Particle filters (PFs) have become an essential tool for disease surveillance, as they can estimate hidden epidemic states in nonlinear and non-Gaussian models. In epidemic modelling, population dynamics may be governed by distinct regimes such as endemic or outbreak phases which can be represented using Markov-switching state-space models. In many real-world surveillance systems, data often arrives with delays or in the wrong temporal order, producing out-of-sequence (OOS) measurements that pertain to past time points rather than the current one. While existing PF methods can incorporate OOS measurements through particle reweighting, these approaches are limited in their ability to fully adjust past latent trajectories. To address this, we introduce a Markov-switching fixed-lag particle filter (FL-PF) that resimulates particle trajectories within a user-specified lag window, allowing OOS measurements to retroactively update both state and model estimates. By explicitly reevaluating historical samples, the FL-PF improves the accuracy and timeliness of outbreak detection and reduces false alarms. We also show how to compute the log-likelihood within the FL-PF framework, enabling parameter estimation using Sequential Monte Carlo squared (SMC$^2$). Together, these contributions extend the applicability of PFs to surveillance systems where retrospective data are common, offering a more robust framework for monitoring disease outbreaks and parameter inference.

Post-training quantization (PTQ) aims to preserve model-level behavior; however, most methods focus on individual linear layers. Even recent extensions, such as QEP and LoaQ, which mitigate error propagation or target specific submodules, still rely on layer-wise formulations and fail to capture the behavior of larger submodules. We introduce Layer-Projected Coordinate Descent (LPCD), a unified framework that extends PTQ beyond layers by optimizing relaxed objectives across arbitrary submodules and projecting the solutions with layer-wise quantizers. LPCD generalizes existing methods and provides a principled approach to quantizing complex submodules while maintaining the efficiency and compatibility of layer-wise PTQ pipelines. Across diverse LLM architectures and bit-widths, LPCD-based submodule quantization consistently enhances both layer-wise PTQ methods and existing submodule approaches.

Understanding the temporal dynamics of functional brain connectivity is important for addressing various questions in network neuroscience, such as how connectivity affects cognition and changes with disease. A fundamental challenge is to evaluate whether connectivity truly exhibits dynamics, or simply is static. The most common approach uses sliding-window methods to model functional connectivity over time, and this is often combined with frequentist hypothesis testing frameworks to evaluate dynamics. However, this requires defining appropriate sampling distributions and hyperparameters, such as window length, which imposes specific assumptions on the dynamics. Here, we explore how these assumptions influence the detection of dynamic connectivity, and introduce an alternative approach based on Bayesian hypothesis testing with Wishart processes. This framework estimates uncertainty in the connectivity estimates, and uses this to provide strength of evidence for both dynamic and static connectivity. It encodes assumptions through prior distributions, allowing prior knowledge on the time-dependent structure of connectivity to be incorporated into the model. Using simulations, we compare the two approaches and demonstrate how different assumptions affect the detection of dynamic connectivity. Finally, by applying both approaches to an fMRI working-memory task, we find that conclusions at the group-level increase robustness to modeling choices. Our work highlights the importance of carefully considering modeling assumptions when evaluating dynamic connectivity.

Environmental exposures, such as air pollution and extreme temperatures, have complex effects on human health. These effects are often characterized by non-linear exposure-lag-response relationships and delayed impacts over time. Accurately capturing these dynamics is crucial for informing public health interventions. The Distributed Lag Non-Linear Model (DLNM) is a flexible statistical framework for estimating such effects in epidemiological research. However, standard DLNM implementations typically assume a homogeneous exposure-lag-response association across the study region, overlooking potential spatial heterogeneity, which can lead to biased risk estimates. To address this limitation, we introduce DLNM-Clust: a novel mixture of DLNMs that extends the traditional DLNM. Within a Bayesian framework, DLNM-Clust probabilistically assigns each geographic unit to one of $C$ latent spatial clusters, each of which is defined by a distinct DLNM specification. This approach allows capturing both common patterns and singular deviations in the exposure-lag-response surface. We demonstrate the method using municipality-level time-series data on the relationship between air pollution and the incidence of COVID-19 in Belgium. Our results emphasize the importance of spatially aware modeling strategies in environmental epidemiology, facilitating region-specific risk assessment and supporting the development of targeted public health initiatives.

This paper introduces a comprehensive unified framework for constructing multi-view diffusion geometries through intertwined multi-view diffusion trajectories (MDTs), a class of inhomogeneous diffusion processes that iteratively combine the random walk operators of multiple data views. Each MDT defines a trajectory-dependent diffusion operator with a clear probabilistic and geometric interpretation, capturing over time the interplay between data views. Our formulation encompasses existing multi-view diffusion models, while providing new degrees of freedom for view interaction and fusion. We establish theoretical properties under mild assumptions, including ergodicity of both the point-wise operator and the process in itself. We also derive MDT-based diffusion distances, and associated embeddings via singular value decompositions. Finally, we propose various strategies for learning MDT operators within the defined operator space, guided by internal quality measures. Beyond enabling flexible model design, MDTs also offer a neutral baseline for evaluating diffusion-based approaches through comparison with randomly selected MDTs. Experiments show the practical impact of the MDT operators in a manifold learning and data clustering context.

We consider the problem of generalization of arbitrarily overparameterized two-layer ReLU Neural Networks with univariate input. Recent work showed that under square loss, flat solutions (motivated by flat / stable minima and Edge of Stability phenomenon) provably cannot overfit, but it remains unclear whether the same phenomenon holds for logistic loss. This is a puzzling open problem because existing work on logistic loss shows that gradient descent with increasing step size converges to interpolating solutions (at infinity, for the margin-separable cases). In this paper, we prove that the flatness implied generalization is more delicate under logistic loss. On the positive side, we show that flat solutions enjoy near-optimal generalization bounds within a region between the left-most and right-most uncertain sets determined by each candidate solution. On the negative side, we show that there exist arbitrarily flat yet overfitting solutions at infinity that are (falsely) certain everywhere, thus certifying that flatness alone is insufficient for generalization in general. We demonstrate the effects predicted by our theory in a well-controlled simulation study.

Clustering functional data is a challenging task due to intrinsic infinite-dimensionality and the need for stable, data-adaptive partitioning. In this work, we propose a clustering framework based on Random Projections, which simultaneously performs dimensionality reduction and generates multiple stochastic representations of the original functions. Each projection is clustered independently, and the resulting partitions are then aggregated through an ensemble consensus procedure, enhancing robustness and mitigating the influence of any single projection. To focus on the most informative representations, projections are ranked according to clustering quality criteria, and only a selected subset is retained. In particular, we adopt Gaussian Mixture Models as base clusterers and employ the Kullback-Leibler divergence to order the random projections; these choices enable fast computation and eliminate the need to specify the number of clusters a priori. The performance of the proposed methodology is assessed through an extensive simulation study and two real-data applications, one from spectroscopy data for food authentication and one from log-periodograms of speech recording; the obtained results suggest that the proposal represents an effective tool for the clustering of functional data.

Event-driven systems in fields such as neuroscience, social networks, and finance often exhibit dynamics influenced by continuously evolving external covariates. Motivated by these applications, we introduce a new class of multivariate Hawkes processes, in which the spontaneous rate of events is modulated by a diffusion process. This framework allows the point process to adapt dynamically to continuously evolving covariates, capturing both intrinsic self-excitation and external influences. In this article, we establish the probabilistic properties of the coupled process, proving stability and ergodicity under moderate assumptions. Classical functional results, including law of large numbers and mixing properties, are extended to this diffusion-driven setting. Building on these results, we study parametric inference for the Hawkes component: we derive consistency and asymptotic normality of the maximum likelihood estimator in the long-time regime, and derive stronger convergence results under additional assumptions on the covariate process. We further propose hypothesis testing procedures to assess the statistical relevance of the covariate. Simulation studies illustrate the validity of the asymptotic results and the effectiveness of the proposed inference methods. Overall, this work provides theoretical and practical foundations for diffusion-driven Hawkes models.

Discrete event systems are present both in observations of nature, socio economical sciences, and industrial systems. Standard analysis approaches do not usually exploit their dual event / state nature: signals are either modeled as transition event sequences, emphasizing event order alignment, or as categorical or ordinal state timeseries, usually resampled a distorting and costly operation as the observation period and number of events grow. In this work we define state transition event timeseries (STE-ts) and propose a new Selective Temporal Hamming distance (STH) leveraging both transition time and duration-in-state, avoiding costly and distorting resampling on large databases. STH generalizes both resampled Hamming and Jaccard metrics with better precision and computation time, and an ability to focus on multiple states of interest. We validate these benefits on simulated and real-world datasets.

In large-scale hypothesis testing, computing exact $p$-values or $e$-values is often resource-intensive, creating a need for budget-aware inferential methods. We propose a general framework for active hypothesis testing that leverages inexpensive auxiliary statistics to allocate a global computational budget. For each hypothesis, our data-adaptive procedure probabilistically decides whether to compute the exact test statistic or a transformed proxy, guaranteeing a valid $p$-value or $e$-value under the stated assumptions while satisfying the exact budget constraint. We establish optimality for active $e$-values and develop optimality and admissibility theory for active $p$-values. Empirical results from simulations and two real-world applications, including a large-scale genome-wide association study and {a clinical prediction task using a large language model to impute a costly feature}, demonstrate that our framework improves statistical efficiency under fixed resource limits.

We revisit Merton's continuous-time portfolio selection through a data-driven, distributionally robust lens. Our aim is to tap the benefits of frequent trading over short horizons while acknowledging that drift is hard to pin down, whereas volatility can be screened using realized or implied measures for appropriately selected assets. Rather than time-rectangular distributional robust control – which replenishes adversarial power at every instant and induces over-pessimism – we place a single ambiguity set on the drift prior within a Bayesian Merton model. This prior-level ambiguity preserves learning and tractability: a minimax swap reduces the robust control to optimizing a nonlinear functional of the prior, enabling Karatzas and Zhao \cite{KZ98}-type's closed-form evaluation for each candidate prior. We then characterize small-radius worst-case priors under Wasserstein uncertainty via an explicit asymptotically optimal pushforward of the nominal prior, and we calibrate the ambiguity radius through a nonlinear Wasserstein projection tailored to the Merton functional. Synthetic and real-data studies demonstrate reduced pessimism relative to DRC and improved performance over myopic DRO-Markowitz under frequent rebalancing.

Conformal regression provides finite-sample marginal coverage, but interval efficiency depends on how uncertainty is allocated across inputs. We propose Conformal Laplace-Aware Predictive Scaling (CLAPS), which combines a learned heteroscedastic scale with last-layer Laplace uncertainty. The learned variance also weights the training features used to construct the posterior, yielding a noise-weighted support correction in the learned representation. The correction contracts as weighted feature information increases and becomes most influential under localized support imbalance. Experiments show near-nominal marginal coverage and modest efficiency improvements over the matched aleatoric baseline on real regression benchmarks.