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Generative models, such as Generative Adversarial Networks (GANs), have been used for unsupervised anomaly detection. While performance keeps improving, several limitations exist particularly attributed to difficulties at capturing multimodal supports and to the ability to approximate the underlying distribution closer to the tails, i.e. the boundary of the distribution's support. This paper proposes an approach that attempts to alleviate such shortcomings. We propose an invertible-residual-network-based model, the Boundary of Distribution Support Generator (BDSG). GANs generally do not guarantee the existence of a probability distribution and here, we use the recently developed Invertible Residual Network (IResNet) and Residual Flow (ResFlow), for density estimation. These models have not yet been used for anomaly detection. We leverage IResNet and ResFlow for Out-of-Distribution (OoD) sample detection and for sample generation on the boundary using a compound loss function that forces the samples to lie on the boundary. The BDSG addresses non-convex support, disjoint components, and multimodal distributions. Results on synthetic data and data from multimodal distributions, such as MNIST and CIFAR-10, demonstrate competitive performance compared to methods from the literature.

Deep Neural Networks (DNNs), despite their tremendous success in recent years, could still cast doubts on their predictions due to the intrinsic uncertainty associated with their learning process. Ensemble techniques and post-hoc calibrations are two types of approaches that have individually shown promise in improving the uncertainty calibration of DNNs. However, the synergistic effect of the two types of methods has not been well explored. In this paper, we propose a truth discovery framework to integrate ensemble-based and post-hoc calibration methods. Using the geometric variance of the ensemble candidates as a good indicator for sample uncertainty, we design an accuracy-preserving truth estimator with provably no accuracy drop. Furthermore, we show that post-hoc calibration can also be enhanced by truth discovery-regularized optimization. On large-scale datasets including CIFAR and ImageNet, our method shows consistent improvement against state-of-the-art calibration approaches on both histogram-based and kernel density-based evaluation metrics. Our codes are available at //github.com/horsepurve/truly-uncertain.

When the distribution of treatment effect modifiers differs between the trial sample and target population, inverse probability weighting (IPSW) can be applied to achieve an unbiased estimate of the population average treatment effect in the target population. The statistical validity of IPSW is threatened when there are missing data in the target population, including potential missingness in trial sample. However, missing data methods have not been adequately discussed in the current literature. We conducted a set of simulation studies to determine how to apply multiple imputation (MI) in the context of IPSW. We specifically addressed questions such as which variables to include in the imputation model and whether they should come from trial or non-trial portion of the target population. Based on our findings, we recommend including all potential effect modifiers and trial indicator from both trial and non-trial populations, as well as treatment and outcome variables from trial sample in the imputation model as main effects. Additionally, we have illustrated ideas by transporting findings from the Frequent Hemodialysis Network (FHN) Daily Trial to the United States Renal Stage System (USRDS) population.

We study frequentist risk properties of predictive density estimators for mean mixtures of multivariate normal distributions, involving an unknown location parameter $\theta \in \mathbb{R}^d$, and which include multivariate skew normal distributions. We provide explicit representations for Bayesian posterior and predictive densities, including the benchmark minimum risk equivariant (MRE) density, which is minimax and generalized Bayes with respect to an improper uniform density for $\theta$. For four dimensions or more, we obtain Bayesian densities that improve uniformly on the MRE density under Kullback-Leibler loss. We also provide plug-in type improvements, investigate implications for certain type of parametric restrictions on $\theta$, and illustrate and comment the findings based on numerical evaluations.

Estimation of causal effects using machine learning methods has become an active research field in econometrics. In this paper, we study the finite sample performance of meta-learners for estimation of heterogeneous treatment effects under the usage of sample-splitting and cross-fitting to reduce the overfitting bias. In both synthetic and semi-synthetic simulations we find that the performance of the meta-learners in finite samples greatly depends on the estimation procedure. The results imply that sample-splitting and cross-fitting are beneficial in large samples for bias reduction and efficiency of the meta-learners, respectively, whereas full-sample estimation is preferable in small samples. Furthermore, we derive practical recommendations for application of specific meta-learners in empirical studies depending on particular data characteristics such as treatment shares and sample size.

The issues of bias-correction and robustness are crucial in the strategy of divide-and-conquer (DC), especially for asymmetric nonparametric models with massive data. It is known that quantile-based methods can achieve the robustness, but the quantile estimation for nonparametric regression has non-ignorable bias when the error distribution is asymmetric. This paper explores a global bias-corrected DC by quantile-matched composite for nonparametric regressions with general error distributions. The proposed strategies can achieve the bias-correction and robustness, simultaneously. Unlike common DC quantile estimations that use an identical quantile level to construct a local estimator by each local machine, in the new methodologies, the local estimators are obtained at various quantile levels for different data batches, and then the global estimator is elaborately constructed as a weighted sum of the local estimators. In the weighted sum, the weights and quantile levels are well-matched such that the bias of the global estimator is corrected significantly, especially for the case where the error distribution is asymmetric. Based on the asymptotic properties of the global estimator, the optimal weights are attained, and the corresponding algorithms are then suggested. The behaviors of the new methods are further illustrated by various numerical examples from simulation experiments and real data analyses. Compared with the competitors, the new methods have the favorable features of estimation accuracy, robustness, applicability and computational efficiency.

We consider the problem of causal discovery (structure learning) from heterogeneous observational data. Most existing methods assume a homogeneous sampling scheme, which leads to misleading conclusions when violated in many applications. To this end, we propose a novel approach that exploits data heterogeneity to infer possibly cyclic causal structures from causally insufficient systems. The core idea is to model the direct causal effects as functions of exogenous covariates that properly explain data heterogeneity. We investigate structure identifiability properties of the proposed model. Structure learning is carried out in a fully Bayesian fashion, which provides natural uncertainty quantification. We demonstrate its utility through extensive simulations and a real-world application.

This PhD thesis contains several contributions to the field of statistical causal modeling. Statistical causal models are statistical models embedded with causal assumptions that allow for the inference and reasoning about the behavior of stochastic systems affected by external manipulation (interventions). This thesis contributes to the research areas concerning the estimation of causal effects, causal structure learning, and distributionally robust (out-of-distribution generalizing) prediction methods. We present novel and consistent linear and non-linear causal effects estimators in instrumental variable settings that employ data-dependent mean squared prediction error regularization. Our proposed estimators show, in certain settings, mean squared error improvements compared to both canonical and state-of-the-art estimators. We show that recent research on distributionally robust prediction methods has connections to well-studied estimators from econometrics. This connection leads us to prove that general K-class estimators possess distributional robustness properties. We, furthermore, propose a general framework for distributional robustness with respect to intervention-induced distributions. In this framework, we derive sufficient conditions for the identifiability of distributionally robust prediction methods and present impossibility results that show the necessity of several of these conditions. We present a new structure learning method applicable in additive noise models with directed trees as causal graphs. We prove consistency in a vanishing identifiability setup and provide a method for testing substructure hypotheses with asymptotic family-wise error control that remains valid post-selection. Finally, we present heuristic ideas for learning summary graphs of nonlinear time-series models.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.

Class labels have been empirically shown useful in improving the sample quality of generative adversarial nets (GANs). In this paper, we mathematically study the properties of the current variants of GANs that make use of class label information. With class aware gradient and cross-entropy decomposition, we reveal how class labels and associated losses influence GAN's training. Based on that, we propose Activation Maximization Generative Adversarial Networks (AM-GAN) as an advanced solution. Comprehensive experiments have been conducted to validate our analysis and evaluate the effectiveness of our solution, where AM-GAN outperforms other strong baselines and achieves state-of-the-art Inception Score (8.91) on CIFAR-10. In addition, we demonstrate that, with the Inception ImageNet classifier, Inception Score mainly tracks the diversity of the generator, and there is, however, no reliable evidence that it can reflect the true sample quality. We thus propose a new metric, called AM Score, to provide more accurate estimation on the sample quality. Our proposed model also outperforms the baseline methods in the new metric.