Conditional independence testing is a key problem required by many machine learning and statistics tools. In particular, it is one way of evaluating the usefulness of some features on a supervised prediction problem. We propose a novel conditional independence test in a predictive setting, and show that it achieves better power than competing approaches in several settings. Our approach consists in deriving a p-value using a permutation test where the predictive power using the unpermuted dataset is compared with the predictive power of using dataset where the feature(s) of interest are permuted. We conclude that the method achives sensible results on simulated and real datasets.
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Conditional independence testing is an important problem, especially in Bayesian network learning and causal discovery. Due to the curse of dimensionality, testing for conditional independence of continuous variables is particularly challenging. We propose a Kernel-based Conditional Independence test (KCI-test), by constructing an appropriate test statistic and deriving its asymptotic distribution under the null hypothesis of conditional independence. The proposed method is computationally efficient and easy to implement. Experimental results show that it outperforms other methods, especially …
CREST（法国 Crest 研究中心）；ENSAE（法国国家统计与经济管理学院）；Facebook AI Research（脸书人工智能研究院）；Université Paris-Dauphine（巴黎多芬大学）；Technion（以色列理工学院）
We propose a new conditional dependence measure and a statistical test for conditional independence. The measure is based on the difference between analytic kernel embeddings of two well-suited distributions evaluated at a finite set of locations. We obtain its asymptotic distribution under the null hypothesis of conditional independence and design a consistent statistical test from it. We conduct a series of experiments showing that our new test outperforms state-of-the-art methods both in terms of type-I and type-II errors even in the high dimensional setting.
From Conditional to Unconditional Independence: Testing Conditional Independence via Transport Maps
Institute of Statistics and Big Data, Renmin University of China（中国人民大学统计与大数据研究院）；Department of Data Science and Artificial Intelligence, The Hong Kong Polytechnic University（香港理工大学数据科学与人工智能学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
We formulate and analyze a graphical model selection method for inferring the conditional independence graph of a high-dimensional nonstationary Gaussian random process (time series) from a finite-length observation. The observed process samples are assumed uncorrelated over time and having a time-varying marginal distribution. The selection method is based on testing conditional variances obtained for small subsets of process components. This allows to cope with the high-dimensional regime, where the sample size can be (drastically) smaller than the process dimension. We characterize the requ…
Detecting conditional independencies plays a key role in several statistical and machine learning tasks, especially in causal discovery algorithms. In this study, we introduce LCIT (Latent representation based Conditional Independence Test)-a novel non-parametric method for conditional independence testing based on representation learning. Our main contribution involves proposing a generative framework in which to test for the independence between X and Y given Z, we first learn to infer the latent representations of target variables X and Y that contain no information about the conditioning v…
Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；Saarland University（萨尔兰大学）；CISPA Helmholtz Center for Information Security（亥姆霍兹信息安全研究中心CISPA）
Testing for conditional independence is a core aspect of constraint-based causal discovery. Although commonly used tests are perfect in theory, they often fail to reject independence in practice, especially when conditioning on multiple variables. We focus on discrete data and propose a new test based on the notion of algorithmic independence that we instantiate using stochastic complexity. Amongst others, we show that our proposed test, SCI, is an asymptotically unbiased as well as $L_2$ consistent estimator for conditional mutual information (CMI). Further, we show that SCI can be reformulat…
It is often stated in papers tackling the task of inferring Bayesian network structures from data that there are these two distinct approaches: (i) Apply conditional independence tests when testing for the presence or otherwise of edges; (ii) Search the model space using a scoring metric. Here I argue that for complete data and a given node ordering this division is a myth, by showing that cross entropy methods for checking conditional independence are mathematically identical to methods based upon discriminating between models by their overall goodness-of-fit logarithmic scores.
Testing (conditional) independence of multivariate random variables is a task central to statistical inference and modelling in general - though unfortunately one for which to date there does not exist a practicable workflow. State-of-art workflows suffer from the need for heuristic or subjective manual choices, high computational complexity, or strong parametric assumptions. We address these problems by establishing a theoretical link between multivariate/conditional independence testing, and model comparison in the multivariate predictive modelling aka supervised learning task. This link all…
Testing Conditional Mean Independence Using Generative Neural Networks
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Maryland, College Park（马里兰大学帕克分校）；Washington University in St Louis（圣路易斯华盛顿大学）
University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）；University of California Los Angeles（加州大学洛杉矶分校）
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 …
Sequential Kernel-based Conditional Independence Testing via Adaptive Betting
Department of Computer Science, University of British Columbia, Vancouver, Canada（不列颠哥伦比亚大学计算机科学系，温哥华，加拿大）；Alberta Machine Intelligence Institute, Edmonton, Canada（阿尔伯塔机器智能研究所，爱德蒙顿，加拿大）
Measuring conditional independence is one of the important tasks in statistical inference and is fundamental in causal discovery, feature selection, dimensionality reduction, Bayesian network learning, and others. In this work, we explore the connection between conditional independence measures induced by distances on a metric space and reproducing kernels associated with a reproducing kernel Hilbert space (RKHS). For certain distance and kernel pairs, we show the distance-based conditional independence measures to be equivalent to that of kernel-based measures. On the other hand, we also show…
This paper introduces the \textit{weighted partial copula} function for testing conditional independence. The proposed test procedure results from these two ingredients: (i) the test statistic is an explicit Cramer-von Mises transformation of the \textit{weighted partial copula}, (ii) the regions of rejection are computed using a bootstrap procedure which mimics conditional independence by generating samples from the product measure of the estimated conditional marginals. Under conditional independence, the weak convergence of the \textit{weighted partial copula proces}s is established when th…
Meta-Dependence in Conditional Independence Testing
Thayer School of Engineering Dartmouth College（达特茅斯学院工程学院）；Schmidt Center Broad Institute of MIT and Harvard（MIT和哈佛大学Broad研究所Schmidt中心）；Laboratory for Information and Decision Systems Massachusetts Institute of Technology（信息与决策系统实验室麻省理工学院）；Broad Institute of MIT and Harvard（MIT和哈佛大学Broad研究所）
We present and evaluate the Fast (conditional) Independence Test (FIT) -- a nonparametric conditional independence test. The test is based on the idea that when $P(X \mid Y, Z) = P(X \mid Y)$, $Z$ is not useful as a feature to predict $X$, as long as $Y$ is also a regressor. On the contrary, if $P(X \mid Y, Z) \neq P(X \mid Y)$, $Z$ might improve prediction results. FIT applies to thousand-dimensional random variables with a hundred thousand samples in a fraction of the time required by alternative methods. We provide an extensive evaluation that compares FIT to six extant nonparametric indepe…
This article introduces a Bayesian nonparametric method for quantifying the relative evidence in a dataset in favour of the dependence or independence of two variables conditional on a third. The approach uses Polya tree priors on spaces of conditional probability densities, accounting for uncertainty in the form of the underlying distributions in a nonparametric way. The Bayesian perspective provides an inherently symmetric probability measure of conditional dependence or independence, a feature particularly advantageous in causal discovery and not employed in existing procedures of this type…
We show how to estimate a model's test error from unlabeled data, on distributions very different from the training distribution, while assuming only that certain conditional independencies are preserved between train and test. We do not need to assume that the optimal predictor is the same between train and test, or that the true distribution lies in any parametric family. We can also efficiently differentiate the error estimate to perform unsupervised discriminative learning. Our technical tool is the method of moments, which allows us to exploit conditional independencies in the absence of …
Constraint-based causal discovery (CCD) algorithms require fast and accurate conditional independence (CI) testing. The Kernel Conditional Independence Test (KCIT) is currently one of the most popular CI tests in the non-parametric setting, but many investigators cannot use KCIT with large datasets because the test scales cubicly with sample size. We therefore devise two relaxations called the Randomized Conditional Independence Test (RCIT) and the Randomized conditional Correlation Test (RCoT) which both approximate KCIT by utilizing random Fourier features. In practice, both of the proposed …
Fairness via Independence: A (Conditional) Distance Covariance Framework
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, China（华中科技大学数学与统计学院）；Institute of Interdisciplinary Research for Mathematics and Applied Science & Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei, China（华中科技大学数学与应用科学交叉研究 institute 及湖北省工程建模与科学计算重点实验室）
Conditional independence tests (CI tests) have received special attention lately in Machine Learning and Computational Intelligence related literature as an important indicator of the relationship among the variables used by their models. In the field of Probabilistic Graphical Models (PGM)--which includes Bayesian Networks (BN) models--CI tests are especially important for the task of learning the PGM structure from data. In this paper, we propose the Full Bayesian Significance Test (FBST) for tests of conditional independence for discrete datasets. FBST is a powerful Bayesian test for precis…
On the Number of Conditional Independence Tests in Constraint-based Causal Discovery
Eric and Wendy Schmidt Center, Broad Institute of MIT and Harvard（埃里克和文黛斯中心，MIT和哈佛Broad研究所）；Centre de Formació Interdisciplinària Superior, Universitat Politècnica de Catalunya（跨学科高级培训中心，加泰罗尼亚理工大学）；Laboratory for Information and Decision Systems, Massachusetts Institute of Technology（信息与决策系统实验室，麻省理工学院）
This work investigates the intersection property of conditional independence. It states that for random variables $A,B,C$ and $X$ we have that $X$ independent of $A$ given $B,C$ and $X$ independent of $B$ given $A,C$ implies $X$ independent of $(A,B)$ given $C$. Under the assumption that the joint distribution has a continuous density, we provide necessary and sufficient conditions under which the intersection property holds. The result has direct applications to causal inference: it leads to strictly weaker conditions under which the graphical structure becomes identifiable from the joint dis…
We introduce a nonparametric graphical model for discrete node variables based on additive conditional independence. Additive conditional independence is a three way statistical relation that shares similar properties with conditional independence by satisfying the semi-graphoid axioms. Based on this relation we build an additive graphical model for discrete variables that does not suffer from the restriction of a parametric model such as the Ising model. We develop an estimator of the new graphical model via the penalized estimation of the discrete version of the additive precision operator a…
This paper develops a model-free sequential test for conditional independence. The proposed test allows researchers to analyze an incoming i.i.d. data stream with any arbitrary dependency structure, and safely conclude whether a feature is conditionally associated with the response under study. We allow the processing of data points online, as soon as they arrive, and stop data acquisition once significant results are detected, rigorously controlling the type-I error rate. Our test can work with any sophisticated machine learning algorithm to enhance data efficiency to the extent possible. The…
As a crucial problem in statistics is to decide whether additional variables are needed in a regression model. We propose a new multivariate test to investigate the conditional mean independence of Y given X conditioning on some known effect Z, i.e., E(Y|X, Z) = E(Y|Z). Assuming that E(Y|Z) and Z are linearly related, we reformulate an equivalent notion of conditional mean independence through transformation, which is approximated in practice. We apply the martingale difference divergence (Shao and Zhang, 2014) to measure conditional mean dependence, and show that the estimation error from app…
In an independence model, the triplets that represent conditional independences between singletons are called elementary. It is known that the elementary triplets represent the independence model unambiguously under some conditions. In this paper, we show how this representation helps performing some operations with independence models, such as finding the dominant triplets or a minimal independence map of an independence model, or computing the union or intersection of a pair of independence models, or performing causal reasoning. For the latter, we rephrase in terms of conditional independen…
We propose a method for inferring the conditional independence graph (CIG) of a high-dimensional Gaussian vector time series (discrete-time process) from a finite-length observation. By contrast to existing approaches, we do not rely on a parametric process model (such as, e.g., an autoregressive model) for the observed random process. Instead, we only require certain smoothness properties (in the Fourier domain) of the process. The proposed inference scheme works even for sample sizes much smaller than the number of scalar process components if the true underlying CIG is sufficiently sparse. …
University of Amsterdam（阿姆斯特丹大学）；Radboud University Nijmegen（拉德堡德大学）
Real-world complex systems are often modelled by sets of equations with endogenous and exogenous variables. What can we say about the causal and probabilistic aspects of variables that appear in these equations without explicitly solving the equations? We make use of Simon's causal ordering algorithm (Simon, 1953) to construct a causal ordering graph and prove that it expresses the effects of soft and perfect interventions on the equations under certain unique solvability assumptions. We further construct a Markov ordering graph and prove that it encodes conditional independences in the distri…
Conditional independence testing is a fundamental problem underlying causal discovery and a particularly challenging task in the presence of nonlinear and high-dimensional dependencies. Here a fully non-parametric test for continuous data based on conditional mutual information combined with a local permutation scheme is presented. Through a nearest neighbor approach, the test efficiently adapts also to non-smooth distributions due to strongly nonlinear dependencies. Numerical experiments demonstrate that the test reliably simulates the null distribution even for small sample sizes and with hi…
Determinantal point process have recently been used as models in machine learning and this has raised questions regarding the characterizations of conditional independence. In this paper we investigate characterizations of conditional independence. We describe some conditional independencies through the conditions on the kernel of a determinantal point process, and show many can be obtained using the graph induced by a kernel of the $L$-ensemble.
For testing conditional independence (CI) of a response Y and a predictor X given covariates Z, the recently introduced model-X (MX) framework has been the subject of active methodological research, especially in the context of MX knockoffs and their successful application to genome-wide association studies. In this paper, we study the power of MX CI tests, yielding quantitative insights into the role of machine learning and providing evidence in favor of using likelihood-based statistics in practice. Focusing on the conditional randomization test (CRT), we find that its conditional mode of in…
