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We consider a general difference-in-differences model in which the treatment variable of interest may be non-binary and its value may change in each period. It is generally difficult to estimate treatment parameters defined with the potential outcome given the entire path of treatment adoption, because each treatment path may be experienced by only a small number of observations. We propose an alternative approach using the concept of effective treatment, which summarizes the treatment path into an empirically tractable low-dimensional variable, and develop doubly robust identification, estimation, and inference methods. We also provide a companion R software package.

The simultaneous estimation of multiple unknown parameters lies at heart of a broad class of important problems across science and technology. Currently, the state-of-the-art performance in the such problems is achieved by nonparametric empirical Bayes methods. However, these approaches still suffer from two major issues. First, they solve a frequentist problem but do so by following Bayesian reasoning, posing a philosophical dilemma that has contributed to somewhat uneasy attitudes toward empirical Bayes methodology. Second, their computation relies on certain density estimates that become extremely unreliable in some complex simultaneous estimation problems. In this paper, we study these issues in the context of the canonical Gaussian sequence problem. We propose an entirely frequentist alternative to nonparametric empirical Bayes methods by establishing a connection between simultaneous estimation and penalized nonparametric regression. We use flexible regularization strategies, such as shape constraints, to derive accurate estimators without appealing to Bayesian arguments. We prove that our estimators achieve asymptotically optimal regret and show that they are competitive with or can outperform nonparametric empirical Bayes methods in simulations and an analysis of spatially resolved gene expression data.

This paper investigates the asymptotic distribution of the maximum-likelihood estimate (MLE) in multinomial logistic models in the high-dimensional regime where dimension and sample size are of the same order. While classical large-sample theory provides asymptotic normality of the MLE under certain conditions, such classical results are expected to fail in high-dimensions as documented for the binary logistic case in the seminal work of Sur and Cand\`es [2019]. We address this issue in classification problems with 3 or more classes, by developing asymptotic normality and asymptotic chi-square results for the multinomial logistic MLE (also known as cross-entropy minimizer) on null covariates. Our theory leads to a new methodology to test the significance of a given feature. Extensive simulation studies on synthetic data corroborate these asymptotic results and confirm the validity of proposed p-values for testing the significance of a given feature.

Symbolic regression is a machine learning technique that can learn the governing formulas of data and thus has the potential to transform scientific discovery. However, symbolic regression is still limited in the complexity and dimensionality of the systems that it can analyze. Deep learning on the other hand has transformed machine learning in its ability to analyze extremely complex and high-dimensional datasets. We propose a neural network architecture to extend symbolic regression to parametric systems where some coefficient may vary but the structure of the underlying governing equation remains constant. We demonstrate our method on various analytic expressions, ODEs, and PDEs with varying coefficients and show that it extrapolates well outside of the training domain. The neural network-based architecture can also integrate with other deep learning architectures so that it can analyze high-dimensional data while being trained end-to-end. To this end we integrate our architecture with convolutional neural networks to analyze 1D images of varying spring systems.

Regression analysis under the assumption of monotonicity is a well-studied statistical problem and has been used in a wide range of applications. However, there remains a lack of a broadly applicable methodology that permits information borrowing, for efficiency gains, when jointly estimating multiple monotonic regression functions. We introduce such a methodology by extending the isotonic regression problem presented in the article "The isotonic regression problem and its dual" (Barlow and Brunk, 1972). The presented approach can be applied to both fixed and random designs and any number of explanatory variables (regressors). Our framework penalizes pairwise differences in the values (levels) of the monotonic function estimates, with the weight of penalty being determined based on a statistical test, which results in information being shared across data sets if similarities in the regression functions exist. Function estimates are subsequently derived using an iterative optimization routine that uses existing solution algorithms for the isotonic regression problem. Simulation studies for normally and binomially distributed response data illustrate that function estimates are consistently improved if similarities between functions exist, and are not oversmoothed otherwise. We further apply our methodology to analyse two public health data sets: neonatal mortality data for Porto Alegre, Brazil, and stroke patient data for North West England.

Bayesian hierarchical model (BHM) has been widely used in synthesizing information across subgroups. Identifying heterogeneity in the data and determining proper strength of borrow have long been central goals pursued by researchers. Because these two goals are interconnected, we must consider them together. This joint consideration presents two fundamental challenges: (1) How can we balance the trade-off between homogeneity within the cluster and information gain through borrowing? (2) How can we determine the borrowing strength dynamically in different clusters? To tackle challenges, first, we develop a theoretical framework for heterogeneity identification and dynamic information borrowing in BHM. Then, we propose two novel overlapping indices: the overlapping clustering index (OCI) for identifying the optimal clustering result and the overlapping borrowing index (OBI) for assigning proper borrowing strength to clusters. By incorporating these indices, we develop a new method BHMOI (Bayesian hierarchical model with overlapping indices). BHMOI includes a novel weighted K-Means clustering algorithm by maximizing OCI to obtain optimal clustering results, and embedding OBI into BHM for dynamically borrowing within clusters. BHMOI can achieve efficient and robust information borrowing with desirable properties. Examples and simulation studies are provided to demonstrate the effectiveness of BHMOI in heterogeneity identification and dynamic information borrowing.

In recent years, large language models (LLMs) have achieved strong performance on benchmark tasks, especially in zero or few-shot settings. However, these benchmarks often do not adequately address the challenges posed in the real-world, such as that of hierarchical classification. In order to address this challenge, we propose refactoring conventional tasks on hierarchical datasets into a more indicative long-tail prediction task. We observe LLMs are more prone to failure in these cases. To address these limitations, we propose the use of entailment-contradiction prediction in conjunction with LLMs, which allows for strong performance in a strict zero-shot setting. Importantly, our method does not require any parameter updates, a resource-intensive process and achieves strong performance across multiple datasets.

The shocks which hit macroeconomic models such as Vector Autoregressions (VARs) have the potential to be non-Gaussian, exhibiting asymmetries and fat tails. This consideration motivates the VAR developed in this paper which uses a Dirichlet process mixture (DPM) to model the shocks. However, we do not follow the obvious strategy of simply modeling the VAR errors with a DPM since this would lead to computationally infeasible Bayesian inference in larger VARs and potentially a sensitivity to the way the variables are ordered in the VAR. Instead we develop a particular additive error structure inspired by Bayesian nonparametric treatments of random effects in panel data models. We show that this leads to a model which allows for computationally fast and order-invariant inference in large VARs with nonparametric shocks. Our empirical results with nonparametric VARs of various dimensions shows that nonparametric treatment of the VAR errors is particularly useful in periods such as the financial crisis and the pandemic.

In nuclear Thermal Hydraulics (TH) system codes, a significant source of input uncertainty comes from the Physical Model Parameters (PMPs), and accurate uncertainty quantification in these input parameters is crucial for validating nuclear reactor systems within the Best Estimate Plus Uncertainty (BEPU) framework. Inverse Uncertainty Quantification (IUQ) method has been used to quantify the uncertainty of PMPs from a Bayesian perspective. This paper introduces a novel hierarchical Bayesian model for IUQ which aims to mitigate two existing challenges: the high variability of PMPs under varying experimental conditions, and unknown model discrepancies or outliers causing over-fitting issues for the PMPs. The proposed hierarchical model is compared with the conventional single-level Bayesian model based on the PMPs in TRACE using the measured void fraction data in the Boiling Water Reactor Full-size Fine-mesh Bundle Test (BFBT) benchmark. A Hamiltonian Monte Carlo Method - No U-Turn Sampler (NUTS) is used for posterior sampling in the hierarchical structure. The results demonstrate the effectiveness of the proposed hierarchical structure in providing better estimates of the posterior distributions of PMPs and being less prone to over-fitting. The proposed hierarchical model also demonstrates a promising approach for generalizing IUQ to larger databases with a broad range of experimental conditions and different geometric setups.

There is increasing interest in modeling high-dimensional longitudinal outcomes in applications such as developmental neuroimaging research. Growth curve model offers a useful tool to capture both the mean growth pattern across individuals, as well as the dynamic changes of outcomes over time within each individual. However, when the number of outcomes is large, it becomes challenging and often infeasible to tackle the large covariance matrix of the random effects involved in the model. In this article, we propose a high-dimensional response growth curve model, with three novel components: a low-rank factor model structure that substantially reduces the number of parameters in the large covariance matrix, a re-parameterization formulation coupled with a sparsity penalty that selects important fixed and random effect terms, and a computational trick that turns the inversion of a large matrix into the inversion of a stack of small matrices and thus considerably speeds up the computation. We develop an efficient expectation-maximization type estimation algorithm, and demonstrate the competitive performance of the proposed method through both simulations and a longitudinal study of brain structural connectivity in association with human immunodeficiency virus.