Marginal integration for nonparametric causal inference

We consider the problem of inferring the total causal effect of a single variable intervention on a (response) variable of interest. We propose a certain marginal integration regression technique for a very general class of potentially nonlinear structural equation models (SEMs) with known structure, or at least known superset of adjustment variables: we call the procedure S-mint regression. We easily derive that it achieves the convergence rate as for nonparametric regression: for example, single variable intervention effects can be estimated with convergence rate $n^{-2/5}$ assuming smoothness with twice differentiable functions. Our result can also be seen as a major robustness property with respect to model misspecification which goes much beyond the notion of double robustness. Furthermore, when the structure of the SEM is not known, we can estimate (the equivalence class of) the directed acyclic graph corresponding to the SEM, and then proceed by using S-mint based on these estimates. We empirically compare the S-mint regression method with more classical approaches and argue that the former is indeed more robust, more reliable and substantially simpler.Comment: 40 pages, 14 figure

Nonparametric Independence Screening in Sparse Ultra-High Dimensional Additive Models

A variable screening procedure via correlation learning was proposed Fan and Lv (2008) to reduce dimensionality in sparse ultra-high dimensional models. Even when the true model is linear, the marginal regression can be highly nonlinear. To address this issue, we further extend the correlation learning to marginal nonparametric learning. Our nonparametric independence screening is called NIS, a specific member of the sure independence screening. Several closely related variable screening procedures are proposed. Under the nonparametric additive models, it is shown that under some mild technical conditions, the proposed independence screening methods enjoy a sure screening property. The extent to which the dimensionality can be reduced by independence screening is also explicitly quantified. As a methodological extension, an iterative nonparametric independence screening (INIS) is also proposed to enhance the finite sample performance for fitting sparse additive models. The simulation results and a real data analysis demonstrate that the proposed procedure works well with moderate sample size and large dimension and performs better than competing methods.Comment: 48 page

Sparse Additive Models

We present a new class of methods for high-dimensional nonparametric regression and classification called sparse additive models (SpAM). Our methods combine ideas from sparse linear modeling and additive nonparametric regression. We derive an algorithm for fitting the models that is practical and effective even when the number of covariates is larger than the sample size. SpAM is closely related to the COSSO model of Lin and Zhang (2006), but decouples smoothing and sparsity, enabling the use of arbitrary nonparametric smoothers. An analysis of the theoretical properties of SpAM is given. We also study a greedy estimator that is a nonparametric version of forward stepwise regression. Empirical results on synthetic and real data are presented, showing that SpAM can be effective in fitting sparse nonparametric models in high dimensional data

Invariant Causal Prediction for Nonlinear Models

An important problem in many domains is to predict how a system will respond to interventions. This task is inherently linked to estimating the system's underlying causal structure. To this end, Invariant Causal Prediction (ICP) (Peters et al., 2016) has been proposed which learns a causal model exploiting the invariance of causal relations using data from different environments. When considering linear models, the implementation of ICP is relatively straightforward. However, the nonlinear case is more challenging due to the difficulty of performing nonparametric tests for conditional independence. In this work, we present and evaluate an array of methods for nonlinear and nonparametric versions of ICP for learning the causal parents of given target variables. We find that an approach which first fits a nonlinear model with data pooled over all environments and then tests for differences between the residual distributions across environments is quite robust across a large variety of simulation settings. We call this procedure "invariant residual distribution test". In general, we observe that the performance of all approaches is critically dependent on the true (unknown) causal structure and it becomes challenging to achieve high power if the parental set includes more than two variables. As a real-world example, we consider fertility rate modelling which is central to world population projections. We explore predicting the effect of hypothetical interventions using the accepted models from nonlinear ICP. The results reaffirm the previously observed central causal role of child mortality rates

Bayesian Approximate Kernel Regression with Variable Selection

Nonlinear kernel regression models are often used in statistics and machine learning because they are more accurate than linear models. Variable selection for kernel regression models is a challenge partly because, unlike the linear regression setting, there is no clear concept of an effect size for regression coefficients. In this paper, we propose a novel framework that provides an effect size analog of each explanatory variable for Bayesian kernel regression models when the kernel is shift-invariant --- for example, the Gaussian kernel. We use function analytic properties of shift-invariant reproducing kernel Hilbert spaces (RKHS) to define a linear vector space that: (i) captures nonlinear structure, and (ii) can be projected onto the original explanatory variables. The projection onto the original explanatory variables serves as an analog of effect sizes. The specific function analytic property we use is that shift-invariant kernel functions can be approximated via random Fourier bases. Based on the random Fourier expansion we propose a computationally efficient class of Bayesian approximate kernel regression (BAKR) models for both nonlinear regression and binary classification for which one can compute an analog of effect sizes. We illustrate the utility of BAKR by examining two important problems in statistical genetics: genomic selection (i.e. phenotypic prediction) and association mapping (i.e. inference of significant variants or loci). State-of-the-art methods for genomic selection and association mapping are based on kernel regression and linear models, respectively. BAKR is the first method that is competitive in both settings.Comment: 22 pages, 3 figures, 3 tables; theory added; new simulations presented; references adde

A Selective Review of Group Selection in High-Dimensional Models

Grouping structures arise naturally in many statistical modeling problems. Several methods have been proposed for variable selection that respect grouping structure in variables. Examples include the group LASSO and several concave group selection methods. In this article, we give a selective review of group selection concerning methodological developments, theoretical properties and computational algorithms. We pay particular attention to group selection methods involving concave penalties. We address both group selection and bi-level selection methods. We describe several applications of these methods in nonparametric additive models, semiparametric regression, seemingly unrelated regressions, genomic data analysis and genome wide association studies. We also highlight some issues that require further study.Comment: Published in at http://dx.doi.org/10.1214/12-STS392 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org