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Register a new userTesting Conditional Independence in Supervised Learning Algorithms
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We propose the conditional predictive impact (CPI), a consistent and unbiased estimator of the association between one or several features and a given outcome, conditional on a reduced feature set. Building on the knockoff framework of Cand`es et al. (2018), we develop a novel testing procedure that works in conjunction with any valid knockoff sampler, supervised learning algorithm, and loss function. The CPI can be efficiently computed for high-dimensional data without any sparsity constraints. We demonstrate convergence criteria for the CPI and develop statistical inference procedures for evaluating its magnitude, significance, and precision. These tests aid in feature and model selection, extending traditional frequentist and Bayesian techniques to general supervised learning tasks. The CPI may also be applied in causal discovery to identify underlying multivariate graph structures. We test our method using various algorithms, including linear regression, neural networks, random forests, and support vector machines. Empirical results show that the CPI compares favorably to alternative variable importance measures and other nonparametric tests of conditional independence on a diverse array of real and simulated datasets. Simulations confirm that our inference procedures successfully control Type I error and achieve nominal coverage probability. Our method has been implemented in an R package, cpi, which can be downloaded from https://github.com/dswatson/cpi.
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We consider the hypothesis testing problem of detecting conditional dependence, with a focus on high-dimensional feature spaces. Our contribution is a new test statistic based on samples from a generative adversarial network designed to approximate directly a conditional distribution that encodes the null hypothesis, in a manner that maximizes power (the rate of true negatives). We show that such an approach requires only that density approximation be viable in order to ensure that we control type I error (the rate of false positives); in particular, no assumptions need to be made on the form of the distributions or feature dependencies. Using synthetic simulations with high-dimensional data we demonstrate significant gains in power over competing methods. In addition, we illustrate the use of our test to discover causal markers of disease in genetic data.
In this article, we consider the problem of high-dimensional conditional independence testing, which is a key building block in statistics and machine learning. We propose a double generative adversarial networks (GANs)-based inference procedure. We first introduce a double GANs framework to learn two generators, and integrate the two generators to construct a doubly-robust test statistic. We next consider multiple generalized covariance measures, and take their maximum as our test statistic. Finally, we obtain the empirical distribution of our test statistic through multiplier bootstrap. We show that our test controls type-I error, while the power approaches one asymptotically. More importantly, these theoretical guarantees are obtained under much weaker and practically more feasible conditions compared to existing tests. We demonstrate the efficacy of our test through both synthetic and real datasets.
We consider the problem of conditional independence testing of $X$ and $Y$ given $Z$ where $X,Y$ and $Z$ are three real random variables and $Z$ is continuous. We focus on two main cases - when $X$ and $Y$ are both discrete, and when $X$ and $Y$ are both continuous. In view of recent results on conditional independence testing (Shah and Peters, 2018), one cannot hope to design non-trivial tests, which control the type I error for all absolutely continuous conditionally independent distributions, while still ensuring power against interesting alternatives. Consequently, we identify various, natural smoothness assumptions on the conditional distributions of $X,Y|Z=z$ as $z$ varies in the support of $Z$, and study the hardness of conditional independence testing under these smoothness assumptions. We derive matching lower and upper bounds on the critical radius of separation between the null and alternative hypotheses in the total variation metric. The tests we consider are easily implementable and rely on binning the support of the continuous variable $Z$. To complement these results, we provide a new proof of the hardness result of Shah and Peters.
Identifying statistically significant dependency between variables is a key step in scientific discoveries. Many recent methods, such as distance and kernel tests, have been proposed for valid and consistent independence testing and can be applied to data in Euclidean and non-Euclidean spaces. However, in those works, $n$ pairs of points in $mathcal{X} times mathcal{Y}$ are observed. Here, we consider the setting where a pair of $n times n$ graphs are observed, and the corresponding adjacency matrices are treated as kernel matrices. Under a $rho$-correlated stochastic block model, we demonstrate that a naive test (permutation and Pearsons) for a conditional dependency graph model is invalid. Instead, we propose a block-permutation procedure. We prove that our procedure is valid and consistent -- even when the two graphs have different marginal distributions, are weighted or unweighted, and the latent vertex assignments are unknown -- and provide sufficient conditions for the tests to estimate $rho$. Simulations corroborate these results on both binary and weighted graphs. Applying these tests to the whole-organism, single-cell-resolution structural connectomes of C. elegans, we identify strong statistical dependency between the chemical synapse connectome and the gap junction connectome.
We derive new algorithms for online multiple testing that provably control false discovery exceedance (FDX) while achieving orders of magnitude more power than previous methods. This statistical advance is enabled by the development of new algorithmic ideas: earlier algorithms are more static while our new ones allow for the dynamical adjustment of testing levels based on the amount of wealth the algorithm has accumulated. We demonstrate that our algorithms achieve higher power in a variety of synthetic experiments. We also prove that SupLORD can provide error control for both FDR and FDX, and controls FDR at stopping times. Stopping times are particularly important as they permit the experimenter to end the experiment arbitrarily early while maintaining desired control of the FDR. SupLORD is the first non-trivial algorithm, to our knowledge, that can control FDR at stopping times in the online setting.
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