Many model selection algorithms produce a path of fits specifying a sequence of increasingly complex models. Given such a sequence and the data used to produce them, we consider the problem of choosing the least complex model that is not falsified by the data. Extending the selected-model tests of Fithian et al. (2014), we construct p-values for each step in the path which account for the adaptive selection of the model path using the data. In the case of linear regression, we propose two specific tests, the max-t test for forward stepwise regression (generalizing a proposal of Buja and Brown (2014)), and the next-entry test for the lasso. These tests improve on the power of the saturated-model test of Tibshirani et al. (2014), sometimes dramatically. In addition, our framework extends beyond linear regression to a much more general class of parametric and nonparametric model selection problems. To select a model, we can feed our single-step p-values as inputs into sequential stopping rules such as those proposed by G'Sell et al. (2013) and Li and Barber (2015), achieving control of the familywise error rate or false discovery rate (FDR) as desired. The FDR-controlling rules require the null p-values to be independent of each other and of the non-null p-values, a condition not satisfied by the saturated-model p-values of Tibshirani et al. (2014). We derive intuitive and general sufficient conditions for independence, and show that our proposed constructions yield independent p-values.

Genome-wide association studies have proven to be essential for understanding the genetic basis of disease. However, many complex traits---personality traits, facial features, disease subtyping---are inherently high-dimensional, impeding simple approaches to association mapping. We developed a nonparametric Bayesian reduced rank regression model for multi-SNP, multi-trait association mapping that does not require the rank of the linear subspace to be specified. We show in simulations and real data that our model shares strength over SNPs and over correlated traits, improving statistical power to identify genetic associations with an interpretable, SNP-supervised low-dimensional linear projection of the high-dimensional phenotype. On the HapMap phase 3 gene expression QTL study data, we identify pleiotropic expression QTLs that classical univariate tests are underpowered to find and that two step approaches cannot recover. Our Python software, BERRRI, is publicly available at GitHub: https://github.com/ashlee1031/BERRRI.

This paper proposes a distributionally robust approach to logistic regression. We use the Wasserstein distance to construct a ball in the space of probability distributions centered at the uniform distribution on the training samples. If the radius of this ball is chosen judiciously, we can guarantee that it contains the unknown data-generating distribution with high confidence. We then formulate a distributionally robust logistic regression model that minimizes a worst-case expected logloss function, where the worst case is taken over all distributions in the Wasserstein ball. We prove that this optimization problem admits a tractable reformulation and encapsulates the classical as well as the popular regularized logistic regression problems as special cases. We further propose a distributionally robust approach based on Wasserstein balls to compute upper and lower confidence bounds on the misclassification probability of the resulting classifier. These bounds are given by the optimal values of two highly tractable linear programs. We validate our theoretical out-of-sample guarantees through simulated and empirical experiments.

For the problem of high-dimensional sparse linear regression, it is known that an $\ell_0$-based estimator can achieve a $1/n$ "fast" rate on the prediction error without any conditions on the design matrix, whereas in absence of restrictive conditions on the design matrix, popular polynomial-time methods only guarantee the $1/\sqrt{n}$ "slow" rate. In this paper, we show that the slow rate is intrinsic to a broad class of M-estimators. In particular, for estimators based on minimizing a least-squares cost function together with a (possibly non-convex) coordinate-wise separable regularizer, there is always a "bad" local optimum such that the associated prediction error is lower bounded by a constant multiple of $1/\sqrt{n}$. For convex regularizers, this lower bound applies to all global optima. The theory is applicable to many popular estimators, including convex $\ell_1$-based methods as well as M-estimators based on nonconvex regularizers, including the SCAD penalty or the MCP regularizer. In addition, for a broad class of nonconvex regularizers, we show that the bad local optima are very common, in that a broad class of local minimization algorithms with random initialization will typically converge to a bad solution.

What is the difference of a prediction that is made with a causal model and a non-causal model? Suppose we intervene on the predictor variables or change the whole environment. The predictions from a causal model will in general work as well under interventions as for observational data. In contrast, predictions from a non-causal model can potentially be very wrong if we actively intervene on variables. Here, we propose to exploit this invariance of a prediction under a causal model for causal inference: given different experimental settings (for example various interventions) we collect all models that do show invariance in their predictive accuracy across settings and interventions. The causal model will be a member of this set of models with high probability. This approach yields valid confidence intervals for the causal relationships in quite general scenarios. We examine the example of structural equation models in more detail and provide sufficient assumptions under which the set of causal predictors becomes identifiable. We further investigate robustness properties of our approach under model misspecification and discuss possible extensions. The empirical properties are studied for various data sets, including large-scale gene perturbation experiments.

This paper addresses the problem of active learning of a multi-output Gaussian process (MOGP) model representing multiple types of coexisting correlated environmental phenomena. In contrast to existing works, our active learning problem involves selecting not just the most informative sampling locations to be observed but also the types of measurements at each selected location for minimizing the predictive uncertainty (i.e., posterior joint entropy) of a target phenomenon of interest given a sampling budget. Unfortunately, such an entropy criterion scales poorly in the numbers of candidate sampling locations and selected observations when optimized. To resolve this issue, we first exploit a structure common to sparse MOGP models for deriving a novel active learning criterion. Then, we exploit a relaxed form of submodularity property of our new criterion for devising a polynomial-time approximation algorithm that guarantees a constant-factor approximation of that achieved by the optimal set of selected observations. Empirical evaluation on real-world datasets shows that our proposed approach outperforms existing algorithms for active learning of MOGP and single-output GP models.

Large sample size brings the computation bottleneck for modern data analysis. Subsampling is one of efficient strategies to handle this problem. In previous studies, researchers make more fo- cus on subsampling with replacement (SSR) than on subsampling without replacement (SSWR). In this paper we investigate a kind of SSWR, poisson subsampling (PSS), for fast algorithm in ordinary least-square problem. We establish non-asymptotic property, i.e, the error bound of the correspond- ing subsample estimator, which provide a tradeoff between computation cost and approximation efficiency. Besides the non-asymptotic result, we provide asymptotic consistency and normality of the subsample estimator. Methodologically, we propose a two-step subsampling algorithm, which is efficient with respect to a statistical objective and independent on the linear model assumption.. Synthetic and real data are used to empirically study our proposed subsampling strategies. We argue by these empirical studies that, (1) our proposed two-step algorithm has obvious advantage when the assumed linear model does not accurate, and (2) the PSS strategy performs obviously better than SSR when the subsampling ratio increases.

We propose a likelihood ratio based inferential framework for high dimensional semiparametric generalized linear models. This framework addresses a variety of challenging problems in high dimensional data analysis, including incomplete data, selection bias, and heterogeneous multitask learning. Our work has three main contributions. (i) We develop a regularized statistical chromatography approach to infer the parameter of interest under the proposed semiparametric generalized linear model without the need of estimating the unknown base measure function. (ii) We propose a new framework to construct post-regularization confidence regions and tests for the low dimensional components of high dimensional parameters. Unlike existing post-regularization inferential methods, our approach is based on a novel directional likelihood. In particular, the framework naturally handles generic regularized estimators with nonconvex penalty functions and it can be used to infer least false parameters under misspecified models. (iii) We develop new concentration inequalities and normal approximation results for U-statistics with unbounded kernels, which are of independent interest. We demonstrate the consequences of the general theory by using an example of missing data problem. Extensive simulation studies and real data analysis are provided to illustrate our proposed approach.

A significant hurdle for analyzing large sample data is the lack of effective statistical computing and inference methods. An emerging powerful approach for analyzing large sample data is subsampling, by which one takes a random subsample from the original full sample and uses it as a surrogate for subsequent computation and estimation. In this paper, we study subsampling methods under two scenarios: approximating the full sample ordinary least-square (OLS) estimator and estimating the coefficients in linear regression. We present two algorithms, weighted estimation algorithm and unweighted estimation algorithm, and analyze asymptotic behaviors of their resulting subsample estimators under general conditions. For the weighted estimation algorithm, we propose a criterion for selecting the optimal sampling probability by making use of the asymptotic results. On the basis of the criterion, we provide two novel subsampling methods, the optimal subsampling and the predictor- length subsampling methods. The predictor-length subsampling method is based on the L2 norm of predictors rather than leverage scores. Its computational cost is scalable. For unweighted estimation algorithm, we show that its resulting subsample estimator is not consistent to the full sample OLS estimator. However, it has better performance than the weighted estimation algorithm for estimating the coefficients. Simulation studies and a real data example are used to demonstrate the effectiveness of our proposed subsampling methods.

Additive principal components (APCs for short) are a nonlinear generalization of linear principal components. We focus on smallest APCs to describe additive nonlinear constraints that are approximately satisfied by the data. Thus APCs fit data with implicit equations that treat the variables symmetrically, as opposed to regression analyses which fit data with explicit equations that treat the data asymmetrically by singling out a response variable. We propose a regularized data-analytic procedure for APC estimation using kernel methods. In contrast to existing approaches to APCs that are based on regularization through subspace restriction, kernel methods achieve regularization through shrinkage and therefore grant distinctive flexibility in APC estimation by allowing the use of infinite-dimensional functions spaces for searching APC transformation while retaining computational feasibility. To connect population APCs and kernelized finite-sample APCs, we study kernelized population APCs and their associated eigenproblems, which eventually lead to the establishment of consistency of the estimated APCs. Lastly, we discuss an iterative algorithm for computing kernelized finite-sample APCs.