Logistic regression remains one of the most widely used tools in applied statistics, machine learning and data science. Practical datasets often have a substantial number of features $d$ relative to the sample size $n$. In these cases, the logistic regression maximum likelihood estimator (MLE) is biased, and its standard large-sample approximation is poor. In this paper, we develop an improved method for debiasing predictions and estimating frequentist uncertainty for such datasets. We build on recent work characterizing the asymptotic statistical behavior of the MLE in the regime where the aspect ratio $d / n$, instead of the number of features $d$, remains fixed as $n$ grows. In principle, this approximation facilitates bias and uncertainty corrections, but in practice, these corrections require an estimate of the signal strength of the predictors. Our main contribution is SLOE, an estimator of the signal strength with convergence guarantees that reduces the computation time of estimation and inference by orders of magnitude. The bias correction that this facilitates also reduces the variance of the predictions, yielding narrower confidence intervals with higher (valid) coverage of the true underlying probabilities and parameters. We provide an open source package for this method, available at https://github.com/google-research/sloe-logistic.

When linear models are fit to ill-conditioned or confounded narrow-data, TRACE plots are useful in demonstrating and justifying deliberately biased estimation. This makes TRACE diagnostics powerful "visual" displays. If advanced students of regression are trained in interpretation of Trace plots, they could help admininstrators capable of basic statistical thinking avoid misinterpretations of questionable regression coefficient estimates. All five types of ridge TRACE plots for a wide variety of ridge paths can be explored using R-functions. For example, the RXshrink aug.lars() function generates TRACE s for Least-Angle, Lasso and Forward Stagewise methods (Efron, Hastie, Johnstone and Tibshirani 2004; Hastie and

We study an interesting and challenging problem, learning any part of a Bayesian network (BN) structure. In this challenge, it will be computationally inefficient using existing global BN structure learning algorithms to find an entire BN structure to achieve the part of a BN structure in which we are interested. And local BN structure learning algorithms encounter the false edge orientation problem when they are directly used to tackle this challenging problem. In this paper, we first present a new concept of Expand-Backtracking to explain why local BN structure learning methods have the false edge orientation problem, then propose APSL, an efficient and accurate Any Part of BN Structure Learning algorithm. Specifically, APSL divides the V-structures in a Markov blanket (MB) into two types: collider V-structure and non-collider V-structure, then it starts from a node of interest and recursively finds both collider V-structures and non-collider V-structures in the found MBs, until the part of a BN structure in which we are interested are oriented. To improve the efficiency of APSL, we further design the APSL-FS algorithm using Feature Selection, APSL-FS. Using six benchmark BNs, the extensive experiments have validated the efficiency and accuracy of our methods.

We develop a novel hybrid method for Bayesian network structure learning called partitioned hybrid greedy search (pHGS), composed of three distinct yet compatible new algorithms: Partitioned PC (pPC) accelerates skeleton learning via a divide-and-conquer strategy, $p$-value adjacency thresholding (PATH) effectively accomplishes parameter tuning with a single execution, and hybrid greedy initialization (HGI) maximally utilizes constraint-based information to obtain a high-scoring and well-performing initial graph for greedy search. We establish structure learning consistency of our algorithms in the large-sample limit, and empirically validate our methods individually and collectively through extensive numerical comparisons. The combined merits of pPC and PATH achieve significant computational reductions compared to the PC algorithm without sacrificing the accuracy of estimated structures, and our generally applicable HGI strategy reliably improves the estimation structural accuracy of popular hybrid algorithms with negligible additional computational expense. Our empirical results demonstrate the superior empirical performance of pHGS against many state-of-the-art structure learning algorithms.

Mapping of spatial hotspots, i.e., regions with significantly higher rates or probability density of generating certain events (e.g., disease or crime cases), is a important task in diverse societal domains, including public health, public safety, transportation, agriculture, environmental science, etc. Clustering techniques required by these domains differ from traditional clustering methods due to the high economic and social costs of spurious results (e.g., false alarms of crime clusters). As a result, statistical rigor is needed explicitly to control the rate of spurious detections. To address this challenge, techniques for statistically-robust clustering have been extensively studied by the data mining and statistics communities. In this survey we present an up-to-date and detailed review of the models and algorithms developed by this field. We first present a general taxonomy of the clustering process with statistical rigor, covering key steps of data and statistical modeling, region enumeration and maximization, significance testing, and data update. We further discuss different paradigms and methods within each of key steps. Finally, we highlight research gaps and potential future directions, which may serve as a stepping stone in generating new ideas and thoughts in this growing field and beyond.

In this paper we perform a numerious numerical studies for the problem of low-rank matrix completion. We compare the Bayesain approaches and a recently introduced de-biased estimator which provides a useful way to build confidence intervals of interest. From a theoretical viewpoint, the de-biased estimator comes with a sharp minimax-optinmal rate of estimation error whereas the Bayesian approach reaches this rate with an additional logarithmic factor. Our simulation studies show originally interesting results that the de-biased estimator is just as good as the Bayesain estimators. Moreover, Bayesian approaches are much more stable and can outperform the de-biased estimator in the case of small samples. However, we also find that the length of the confidence intervals revealed by the de-biased estimator for an entry is absolutely shorter than the length of the considered credible interval. These suggest further theoretical studies on the estimation error and the concentration for Bayesian methods as they are being quite limited up to present.

There has been a recent resurgence of interest in explainable artificial intelligence (XAI) that aims to reduce the opaqueness of AI-based decision-making systems, allowing humans to scrutinize and trust them. Prior work in this context has focused on the attribution of responsibility for an algorithm's decisions to its inputs wherein responsibility is typically approached as a purely associational concept. In this paper, we propose a principled causality-based approach for explaining black-box decision-making systems that addresses limitations of existing methods in XAI. At the core of our framework lies probabilistic contrastive counterfactuals, a concept that can be traced back to philosophical, cognitive, and social foundations of theories on how humans generate and select explanations. We show how such counterfactuals can quantify the direct and indirect influences of a variable on decisions made by an algorithm, and provide actionable recourse for individuals negatively affected by the algorithm's decision. Unlike prior work, our system, LEWIS: (1)can compute provably effective explanations and recourse at local, global and contextual levels (2)is designed to work with users with varying levels of background knowledge of the underlying causal model and (3)makes no assumptions about the internals of an algorithmic system except for the availability of its input-output data. We empirically evaluate LEWIS on three real-world datasets and show that it generates human-understandable explanations that improve upon state-of-the-art approaches in XAI, including the popular LIME and SHAP. Experiments on synthetic data further demonstrate the correctness of LEWIS's explanations and the scalability of its recourse algorithm.

We focus on the robustness of neural networks for classification. To permit a fair comparison between methods to achieve robustness, we first introduce a standard based on the mensuration of a classifier's degradation. Then, we propose natural perturbed training to robustify the network. Natural perturbations will be encountered in practice: the difference of two images of the same object may be approximated by an elastic deformation (when they have slightly different viewing angles), by occlusions (when they hide differently behind objects), or by saturation, Gaussian noise etc. Training some fraction of the epochs on random versions of such variations will help the classifier to learn better. We conduct extensive experiments on six datasets of varying sizes and granularity. Natural perturbed learning show better and much faster performance than adversarial training on clean, adversarial as well as natural perturbed images. It even improves general robustness on perturbations not seen during the training. For Cifar-10 and STL-10 natural perturbed training even improves the accuracy for clean data and reaches the state of the art performance. Ablation studies verify the effectiveness of natural perturbed training.

We present a simple algorithm for identifying and correcting real-valued noisy labels from a mixture of clean and corrupted samples using Gaussian process regression. A heteroscedastic noise model is employed, in which additive Gaussian noise terms with independent variances are associated with each and all of the observed labels. Thus, the method effectively applies a sample-specific Tikhonov regularization term, generalizing the uniform regularization prevalent in standard Gaussian process regression. Optimizing the noise model using maximum likelihood estimation leads to the containment of the GPR model's predictive error by the posterior standard deviation in leave-one-out cross-validation. A multiplicative update scheme is proposed for solving the maximum likelihood estimation problem under non-negative constraints. While we provide a proof of monotonic convergence for certain special cases, the multiplicative scheme has empirically demonstrated monotonic convergence behavior in virtually all our numerical experiments. We show that the presented method can pinpoint corrupted samples and lead to better regression models when trained on synthetic and real-world scientific data sets.

Most of existing outlier detection methods assume that the outlier factors (i.e., outlierness scoring measures) of data entities (e.g., feature values and data objects) are Independent and Identically Distributed (IID). This assumption does not hold in real-world applications where the outlierness of different entities is dependent on each other and/or taken from different probability distributions (non-IID). This may lead to the failure of detecting important outliers that are too subtle to be identified without considering the non-IID nature. The issue is even intensified in more challenging contexts, e.g., high-dimensional data with many noisy features. This work introduces a novel outlier detection framework and its two instances to identify outliers in categorical data by capturing non-IID outlier factors. Our approach first defines and incorporates distribution-sensitive outlier factors and their interdependence into a value-value graph-based representation. It then models an outlierness propagation process in the value graph to learn the outlierness of feature values. The learned value outlierness allows for either direct outlier detection or outlying feature selection. The graph representation and mining approach is employed here to well capture the rich non-IID characteristics. Our empirical results on 15 real-world data sets with different levels of data complexities show that (i) the proposed outlier detection methods significantly outperform five state-of-the-art methods at the 95%/99% confidence level, achieving 10%-28% AUC improvement on the 10 most complex data sets; and (ii) the proposed feature selection methods significantly outperform three competing methods in enabling subsequent outlier detection of two different existing detectors.