Independence screening is powerful for variable selection when the number of variables is massive. Commonly used independence screening methods are based on marginal correlations or its variants. When some prior knowledge on a certain important set of variables is available, a natural assessment on the relative importance of the other predictors is their conditional contributions to the response given the known set of variables. This results in conditional sure independence screening (CSIS). CSIS produces a rich family of alternative screening methods by different choices of the conditioning set and can help reduce the number of false positive and false negative selections when covariates are highly correlated. This paper proposes and studies CSIS in generalized linear models. We give conditions under which sure screening is possible and derive an upper bound on the number of selected variables. We also spell out the situation under which CSIS yields model selection consistency and the properties of CSIS when a data-driven conditioning set is used. Moreover, we provide two data-driven methods to select the thresholding parameter of conditional screening. The utility of the procedure is illustrated by simulation studies and analysis of two real datasets.
Heavy-tailed high-dimensional data are commonly encountered in various scientific fields and pose great challenges to modern statistical analysis. A natural procedure to address this problem is to use penalized quantile regression with weighted L1-penalty, called weighted robust Lasso (WR-Lasso), in which weights are introduced to ameliorate the bias problem induced by the L1-penalty. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, we investigate the model selection oracle property and establish the asymptotic normality of the WR-Lasso. We show that only mild conditions on the model error distribution are needed. Our theoretical results also reveal that adaptive choice of the weight vector is essential for the WR-Lasso to enjoy these nice asymptotic properties. To make the WR-Lasso practically feasible, we propose a two-step procedure, called adaptive robust Lasso (AR-Lasso), in which the weight vector in the second step is constructed based on the L1-penalized quantile regression estimate from the first step. This two-step procedure is justified theoretically to possess the oracle property and the asymptotic normality. Numerical studies demonstrate the favorable finite-sample performance of the AR-Lasso.
Despite the popularity of Convolutional Neural Networks (CNN), the problem of uncertainty quantification (UQ) of CNN has been largely overlooked. Lack of efficient UQ tools severely limits the application of CNN in certain areas, such as medicine, where prediction uncertainty is critically important. Among the few existing UQ approaches that have been proposed for deep learning, none of them has theoretical consistency that can guarantee the uncertainty quality. To address this issue, we propose a novel bootstrap based framework for the estimation of prediction uncertainty. The inference procedure we use relies on convexified neural networks to establish the theoretical consistency of bootstrap. Our approach has a significantly less computational load than its competitors, as it relies on warm-starts at each bootstrap that avoids refitting the model from scratch. We further explore a novel transfer learning method so our framework can work on arbitrary neural networks. We experimentally demonstrate our approach has a much better performance compared to other baseline CNNs and state-of-the-art methods on various image datasets.
We propose a sequential learning policy for ranking and selection problems, where we use a non-parametric procedure for estimating the value of a policy. Our estimation approach aggregates over a set of kernel functions in order to achieve a more consistent estimator. Each element in the kernel estimation set uses a dierent bandwidth to achieve better aggregation.The nal estimate uses a weighting scheme with the inverse mean square errors of the kernel estimators as weights. This weighting scheme is shown to be optimal under independent kernel estimators. For choosing the measurement, we employ the knowledge gradient method, a myopic policy that relies on predictive distributions to calculate the optimal sampling point. Our method allows a setting where the beliefs are expected to be correlated but the correlation structure is unknown beforehand. This is an extension of the known knowledge gradient with correlated beliefs. Moreover, the proposed policy is asymptotically optimal.
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