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In this paper the authors study a non-linear elliptic-parabolic system, which is motivated by mathematical models for lithium-ion batteries. One state satisfies a parabolic reaction diffusion equation and the other one an elliptic equation. The goal is to determine several scalar parameters in the coupled model in an optimal manner by utilizing a reliable reduced-order approach based on the reduced basis (RB) method. However, the states are coupled through a strongly non-linear function, and this makes the evaluation of online-efficient error estimates difficult. First the well-posedness of the system is proved. Then a Galerkin finite element and RB discretization are described for the coupled system. To certify the RB scheme hierarchical a-posteriori error estimators are utilized in an adaptive trust-region optimization method. Numerical experiments illustrate good approximation properties and efficiencies by using only a relatively small number of reduced basis functions.

High covariate dimensionality is increasingly occurrent in model estimation, and existing techniques to address this issue typically require sparsity or discrete heterogeneity of the unobservable parameter vector. However, neither restriction may be supported by economic theory in some empirical contexts, leading to severe bias and misleading inference. The clustering-based grouped parameter estimator (GPE) introduced in this paper drops both restrictions in favour of the natural one that the parameter support be compact. GPE exhibits robust large sample properties under standard conditions and accommodates both sparse and non-sparse parameters whose support can be bounded away from zero. Extensive Monte Carlo simulations demonstrate the excellent performance of GPE in terms of bias reduction and size control compared to competing estimators. An empirical application of GPE to estimating price and income elasticities of demand for gasoline highlights its practical utility.

In Causal Bayesian Optimization (CBO), an agent intervenes on an unknown structural causal model to maximize a downstream reward variable. In this paper, we consider the generalization where other agents or external events also intervene on the system, which is key for enabling adaptiveness to non-stationarities such as weather changes, market forces, or adversaries. We formalize this generalization of CBO as Adversarial Causal Bayesian Optimization (ACBO) and introduce the first algorithm for ACBO with bounded regret: Causal Bayesian Optimization with Multiplicative Weights (CBO-MW). Our approach combines a classical online learning strategy with causal modeling of the rewards. To achieve this, it computes optimistic counterfactual reward estimates by propagating uncertainty through the causal graph. We derive regret bounds for CBO-MW that naturally depend on graph-related quantities. We further propose a scalable implementation for the case of combinatorial interventions and submodular rewards. Empirically, CBO-MW outperforms non-causal and non-adversarial Bayesian optimization methods on synthetic environments and environments based on real-word data. Our experiments include a realistic demonstration of how CBO-MW can be used to learn users' demand patterns in a shared mobility system and reposition vehicles in strategic areas.

It is widely believed that a joint factor analysis of item responses and response time (RT) may yield more precise ability scores that are conventionally predicted from responses only. For this purpose, a simple-structure factor model is often preferred as it only requires specifying an additional measurement model for item-level RT while leaving the original item response theory (IRT) model for responses intact. The added speed factor indicated by item-level RT correlates with the ability factor in the IRT model, allowing RT data to carry additional information about respondents' ability. However, parametric simple-structure factor models are often restrictive and fit poorly to empirical data, which prompts under-confidence in the suitablity of a simple factor structure. In the present paper, we analyze the 2015 Programme for International Student Assessment (PISA) mathematics data using a semiparametric simple-structure model. We conclude that a simple factor structure attains a decent fit after further parametric assumptions in the measurement model are sufficiently relaxed. Furthermore, our semiparametric model implies that the association between latent ability and speed/slowness is strong in the population, but the form of association is nonlinear. It follows that scoring based on the fitted model can substantially improve the precision of ability scores.

Distribution data refers to a data set where each sample is represented as a probability distribution, a subject area receiving burgeoning interest in the field of statistics. Although several studies have developed distribution-to-distribution regression models for univariate variables, the multivariate scenario remains under-explored due to technical complexities. In this study, we introduce models for regression from one Gaussian distribution to another, utilizing the Wasserstein metric. These models are constructed using the geometry of the Wasserstein space, which enables the transformation of Gaussian distributions into components of a linear matrix space. Owing to their linear regression frameworks, our models are intuitively understandable, and their implementation is simplified because of the optimal transport problem's analytical solution between Gaussian distributions. We also explore a generalization of our models to encompass non-Gaussian scenarios. We establish the convergence rates of in-sample prediction errors for the empirical risk minimizations in our models. In comparative simulation experiments, our models demonstrate superior performance over a simpler alternative method that transforms Gaussian distributions into matrices. We present an application of our methodology using weather data for illustration purposes.

Typically, statistical graphical models are either continuous and parametric (Gaussian, parameterized by the graph-dependent precision matrix) or discrete and non-parametric (with graph-dependent probabilities of cells). Eventually, the two types are mixed. We propose a way to break this dichotomy by introducing two discrete parametric graphical models on finite decomposable graphs: the graph negative multinomial and the graph multinomial distributions. These models interpolate between the product of univariate negative multinomial and negative multinomial distributions, and between the product of binomial and multinomial distributions, respectively. We derive their Markov decomposition and present probabilistic models leading to both. Additionally, we introduce graphical versions of the Dirichlet distribution and inverted Dirichlet distribution, which serve as conjugate priors for the two discrete graphical Markov models. We derive explicit normalizing constants for both graphical Dirichlet laws and demonstrate that their independence structure (a graphical version of neutrality) yields a strong hyper Markov property for both Bayesian models. We also provide characterization theorems for the generalized Dirichlet distributions via strong hyper Markov property. Finally, we develop a Bayesian model selection procedure for the graphical negative multinomial model with respective Dirichlet-type priors.

Symmetry plays a central role in the sciences, machine learning, and statistics. For situations in which data are known to obey a symmetry, a multitude of methods that exploit symmetry have been developed. Statistical tests for the presence or absence of general group symmetry, however, are largely non-existent. This work formulates non-parametric hypothesis tests, based on a single independent and identically distributed sample, for distributional symmetry under a specified group. We provide a general formulation of tests for symmetry that apply to two broad settings. The first setting tests for the invariance of a marginal or joint distribution under the action of a compact group. Here, an asymptotically unbiased test only requires a computable metric on the space of probability distributions and the ability to sample uniformly random group elements. Building on this, we propose an easy-to-implement conditional Monte Carlo test and prove that it achieves exact $p$-values with finitely many observations and Monte Carlo samples. The second setting tests for the invariance or equivariance of a conditional distribution under the action of a locally compact group. We show that the test for conditional invariance or equivariance can be formulated as particular tests of conditional independence. We implement these tests from both settings using kernel methods and study them empirically on synthetic data. Finally, we apply them to testing for symmetry in geomagnetic satellite data and in two problems from high-energy particle physics.

Modern robotic systems are required to operate in challenging environments, which demand reliable localization under challenging conditions. LiDAR-based localization methods, such as the Iterative Closest Point (ICP) algorithm, can suffer in geometrically uninformative environments that are known to deteriorate point cloud registration performance and push optimization toward divergence along weakly constrained directions. To overcome this issue, this work proposes i) a robust fine-grained localizability detection module, and ii) a localizability-aware constrained ICP optimization module, which couples with the localizability detection module in a unified manner. The proposed localizability detection is achieved by utilizing the correspondences between the scan and the map to analyze the alignment strength against the principal directions of the optimization as part of its fine-grained LiDAR localizability analysis. In the second part, this localizability analysis is then integrated into the scan-to-map point cloud registration to generate drift-free pose updates by enforcing controlled updates or leaving the degenerate directions of the optimization unchanged. The proposed method is thoroughly evaluated and compared to state-of-the-art methods in simulated and real-world experiments, demonstrating the performance and reliability improvement in LiDAR-challenging environments. In all experiments, the proposed framework demonstrates accurate and generalizable localizability detection and robust pose estimation without environment-specific parameter tuning.

Recently developed reduced-order modeling techniques aim to approximate nonlinear dynamical systems on low-dimensional manifolds learned from data. This is an effective approach for modeling dynamics in a post-transient regime where the effects of initial conditions and other disturbances have decayed. However, modeling transient dynamics near an underlying manifold, as needed for real-time control and forecasting applications, is complicated by the effects of fast dynamics and nonnormal sensitivity mechanisms. To begin to address these issues, we introduce a parametric class of nonlinear projections described by constrained autoencoder neural networks in which both the manifold and the projection fibers are learned from data. Our architecture uses invertible activation functions and biorthogonal weight matrices to ensure that the encoder is a left inverse of the decoder. We also introduce new dynamics-aware cost functions that promote learning of oblique projection fibers that account for fast dynamics and nonnormality. To demonstrate these methods and the specific challenges they address, we provide a detailed case study of a three-state model of vortex shedding in the wake of a bluff body immersed in a fluid, which has a two-dimensional slow manifold that can be computed analytically. In anticipation of future applications to high-dimensional systems, we also propose several techniques for constructing computationally efficient reduced-order models using our proposed nonlinear projection framework. This includes a novel sparsity-promoting penalty for the encoder that avoids detrimental weight matrix shrinkage via computation on the Grassmann manifold.

Confounding is a significant obstacle to unbiased estimation of causal effects from observational data. For settings with high-dimensional covariates -- such as text data, genomics, or the behavioral social sciences -- researchers have proposed methods to adjust for confounding by adapting machine learning methods to the goal of causal estimation. However, empirical evaluation of these adjustment methods has been challenging and limited. In this work, we build on a promising empirical evaluation strategy that simplifies evaluation design and uses real data: subsampling randomized controlled trials (RCTs) to create confounded observational datasets while using the average causal effects from the RCTs as ground-truth. We contribute a new sampling algorithm, which we call RCT rejection sampling, and provide theoretical guarantees that causal identification holds in the observational data to allow for valid comparisons to the ground-truth RCT. Using synthetic data, we show our algorithm indeed results in low bias when oracle estimators are evaluated on the confounded samples, which is not always the case for a previously proposed algorithm. In addition to this identification result, we highlight several finite data considerations for evaluation designers who plan to use RCT rejection sampling on their own datasets. As a proof of concept, we implement an example evaluation pipeline and walk through these finite data considerations with a novel, real-world RCT -- which we release publicly -- consisting of approximately 70k observations and text data as high-dimensional covariates. Together, these contributions build towards a broader agenda of improved empirical evaluation for causal estimation.