Heavy tails is a common feature of filtering distributions that results from the nonlinear dynamical and observation processes as well as the uncertainty from physical sensors. In these settings, the Kalman filter and its ensemble version - the ensemble Kalman filter (EnKF) - that have been designed under Gaussian assumptions result in degraded performance. t-distributions are a parametric family of distributions whose tail-heaviness is modulated by a degree of freedom $\nu$. Interestingly, Cauchy and Gaussian distributions correspond to the extreme cases of a t-distribution for $\nu = 1$ and $\nu = \infty$, respectively. Leveraging tools from measure transport (Spantini et al., SIAM Review, 2022), we present a generalization of the EnKF whose prior-to-posterior update leads to exact inference for t-distributions. We demonstrate that this filter is less sensitive to outlying synthetic observations generated by the observation model for small $\nu$. Moreover, it recovers the Kalman filter for $\nu = \infty$. For nonlinear state-space models with heavy-tailed noise, we propose an algorithm to estimate the prior-to-posterior update from samples of joint forecast distribution of the states and observations. We rely on a regularized expectation-maximization (EM) algorithm to estimate the mean, scale matrix, and degree of freedom of heavy-tailed \textit{t}-distributions from limited samples (Finegold and Drton, arXiv preprint, 2014). Leveraging the conditional independence of the joint forecast distribution, we regularize the scale matrix with an $l1$ sparsity-promoting penalization of the log-likelihood at each iteration of the EM algorithm. By sequentially estimating the degree of freedom at each analysis step, our filter can adapt its prior-to-posterior update to the tail-heaviness of the data. We demonstrate the benefits of this new ensemble filter on challenging filtering problems.

In many settings, interventions may be more effective for some individuals than others, so that targeting interventions may be beneficial. We analyze the value of targeting in the context of a large-scale field experiment with over 53,000 college students, where the goal was to use "nudges" to encourage students to renew their financial-aid applications before a non-binding deadline. We begin with baseline approaches to targeting. First, we target based on a causal forest that estimates heterogeneous treatment effects and then assigns students to treatment according to those estimated to have the highest treatment effects. Next, we evaluate two alternative targeting policies, one targeting students with low predicted probability of renewing financial aid in the absence of the treatment, the other targeting those with high probability. The predicted baseline outcome is not the ideal criterion for targeting, nor is it a priori clear whether to prioritize low, high, or intermediate predicted probability. Nonetheless, targeting on low baseline outcomes is common in practice, for example because the relationship between individual characteristics and treatment effects is often difficult or impossible to estimate with historical data. We propose hybrid approaches that incorporate the strengths of both predictive approaches (accurate estimation) and causal approaches (correct criterion); we show that targeting intermediate baseline outcomes is most effective, while targeting based on low baseline outcomes is detrimental. In one year of the experiment, nudging all students improved early filing by an average of 6.4 percentage points over a baseline average of 37% filing, and we estimate that targeting half of the students using our preferred policy attains around 75% of this benefit.

Obtaining continuously updated predictions is a major challenge for personalised medicine. Leveraging combinations of parametric regressions and machine learning approaches, the personalised online super learner (POSL) can achieve such dynamic and personalised predictions. We adapt POSL to predict a repeated continuous outcome dynamically and propose a new way to validate such personalised or dynamic prediction models. We illustrate its performance by predicting the convection volume of patients undergoing hemodiafiltration. POSL outperformed its candidate learners with respect to median absolute error, calibration-in-the-large, discrimination, and net benefit. We finally discuss the choices and challenges underlying the use of POSL.

Multiple data views measured on the same set of participants is becoming more common and has the potential to deepen our understanding of many complex diseases by analyzing these different views simultaneously. Equally important, many of these complex diseases show evidence of subgroup heterogeneity (e.g., by sex or race). HIP (Heterogeneity in Integration and Prediction) is among the first methods proposed to integrate multiple data views while also accounting for subgroup heterogeneity to identify common and subgroup-specific markers of a particular disease. However, HIP is applicable to continuous outcomes and requires programming expertise by the user. Here we propose extensions to HIP that accommodate multi-class, Poisson, and Zero-Inflated Poisson outcomes while retaining the benefits of HIP. Additionally, we introduce an R Shiny application, accessible on shinyapps.io at https://multi-viewlearn.shinyapps.io/HIP_ShinyApp/, that provides an interface with the Python implementation of HIP to allow more researchers to use the method anywhere and on any device. We applied HIP to identify genes and proteins common and specific to males and females that are associated with exacerbation frequency. Although some of the identified genes and proteins show evidence of a relationship with chronic obstructive pulmonary disease (COPD) in existing literature, others may be candidates for future research investigating their relationship with COPD. We demonstrate the use of the Shiny application with a publicly available data. An R-package for HIP would be made available at https://github.com/lasandrall/HIP.

Transformer-based large language models (Vaswani et al., 2017) pretrained with diverse tasks have demonstrated strong ability for in-context learning (ICL), that is, the pretrained models can answer new queries based on a few in-context demonstrations (see, e.g., Brown et al. (2020) and references thereafter). ICL is one of the key abilities contributing to the success of large language models, which allows pretrained models to solve multiple downstream tasks without updating their model parameters. However, the statistical foundation of ICL is still in its infancy. A recent line of research aims to quantify ICL by studying transformers pretrained on the linear regression task with a Gaussian prior (Garg et al., 2022; Akyürek et al., 2022; Li et al., 2023b; Raventós et al., 2023). Specifically, Garg et al. (2022); Akyürek et al. (2022); Li et al. (2023b) study the setting where transformers are pretrained in an online manner using independent linear regression tasks with the same Gaussian prior. They find that such a pretrained transformer can perform ICL on fresh linear regression tasks.

Regression on manifolds, and, more broadly, statistics on manifolds, has garnered significant importance in recent years due to the vast number of applications for this type of data. Circular data is a classic example, but so is data in the space of covariance matrices, data on the Grassmannian manifold obtained as a result of principal component analysis, among many others. In this work we investigate prediction sets for regression scenarios when the response variable, denoted by $Y$, resides in a manifold, and the covariable, denoted by X, lies in Euclidean space. This extends the concepts delineated in [Lei and Wasserman, 2014] to this novel context. Aligning with traditional principles in conformal inference, these prediction sets are distribution-free, indicating that no specific assumptions are imposed on the joint distribution of $(X, Y)$, and they maintain a non-parametric character. We prove the asymptotic almost sure convergence of the empirical version of these regions on the manifold to their population counterparts. The efficiency of this method is shown through a comprehensive simulation study and an analysis involving real-world data.

Many causal estimands are only partially identifiable since they depend on the unobservable joint distribution between potential outcomes. Stratification on pretreatment covariates can yield sharper partial identification bounds; however, unless the covariates are discrete with relatively small support, this approach typically requires consistent estimation of the conditional distributions of the potential outcomes given the covariates. Thus, existing approaches may fail under model misspecification or if consistency assumptions are violated. In this study, we propose a unified and model-agnostic inferential approach for a wide class of partially identified estimands, based on duality theory for optimal transport problems. In randomized experiments, our approach can wrap around any estimates of the conditional distributions and provide uniformly valid inference, even if the initial estimates are arbitrarily inaccurate. Also, our approach is doubly robust in observational studies. Notably, this property allows analysts to use the multiplier bootstrap to select covariates and models without sacrificing validity even if the true model is not included. Furthermore, if the conditional distributions are estimated at semiparametric rates, our approach matches the performance of an oracle with perfect knowledge of the outcome model. Finally, we propose an efficient computational framework, enabling implementation on many practical problems in causal inference.

The growing interest in machine learning problems over graphs with additional node information such as texts, images, or labels has popularized methods that require the costly operation of processing the entire graph. Yet, little effort has been made to the development of fast local methods (i.e. without accessing the entire graph) that extract useful information from such data. To that end, we propose a study of local graph clustering using noisy node labels as a proxy for additional node information. In this setting, nodes receive initial binary labels based on cluster affiliation: 1 if they belong to the target cluster and 0 otherwise. Subsequently, a fraction of these labels is flipped. We investigate the benefits of incorporating noisy labels for local graph clustering. By constructing a weighted graph with such labels, we study the performance of graph diffusion-based local clustering method on both the original and the weighted graphs. From a theoretical perspective, we consider recovering an unknown target cluster with a single seed node in a random graph with independent noisy node labels. We provide sufficient conditions on the label noise under which, with high probability, using diffusion in the weighted graph yields a more accurate recovery of the target cluster. This approach proves more effective than using the given labels alone or using diffusion in the label-free original graph. Empirically, we show that reliable node labels can be obtained with just a few samples from an attributed graph. Moreover, utilizing these labels via diffusion in the weighted graph leads to significantly better local clustering performance across several real-world datasets, improving F1 scores by up to 13%.

Detecting out-of-distribution (OOD) data is a critical challenge in machine learning due to model overconfidence, often without awareness of their epistemological limits. We hypothesize that ``neural collapse'', a phenomenon affecting in-distribution data for models trained beyond loss convergence, also influences OOD data. To benefit from this interplay, we introduce NECO, a novel post-hoc method for OOD detection, which leverages the geometric properties of ``neural collapse'' and of principal component spaces to identify OOD data. Our extensive experiments demonstrate that NECO achieves state-of-the-art results on both small and large-scale OOD detection tasks while exhibiting strong generalization capabilities across different network architectures. Furthermore, we provide a theoretical explanation for the effectiveness of our method in OOD detection. We plan to release the code after the anonymity period.

In the field of global optimization, many existing algorithms face challenges posed by non-convex target functions and high computational complexity or unavailability of gradient information. These limitations, exacerbated by sensitivity to initial conditions, often lead to suboptimal solutions or failed convergence. This is true even for Metaheuristic algorithms designed to amalgamate different optimization techniques to improve their efficiency and robustness. To address these challenges, we develop a sequence of multidimensional integration-based methods that we show to converge to the global optima under some mild regularity conditions. Our probabilistic approach does not require the use of gradients and is underpinned by a mathematically rigorous convergence framework anchored in the nuanced properties of nascent optima distribution. In order to alleviate the problem of multidimensional integration, we develop a latent slice sampler that enjoys a geometric rate of convergence in generating samples from the nascent optima distribution, which is used to approximate the global optima. The proposed Probabilistic Global Optimizer (ProGO) provides a scalable unified framework to approximate the global optima of any continuous function defined on a domain of arbitrary dimension. Empirical illustrations of ProGO across a variety of popular non-convex test functions (having finite global optima) reveal that the proposed algorithm outperforms, by order of magnitude, many existing state-of-the-art methods, including gradient-based, zeroth-order gradient-free, and some Bayesian Optimization methods, in term regret value and speed of convergence. It is, however, to be noted that our approach may not be suitable for functions that are expensive to compute.