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一般混杂下线性非高斯因果模型的参数识别AI 解读

Linear non-Gaussian causal models postulate that each random variable is a linear function of parent variables and non-Gaussian exogenous error terms. We study identification of the linear coefficients when such models contain latent variables. Our focus is on the commonly studied acyclic setting, where each model corresponds to a directed acyclic graph (DAG). For this case, prior literature has demonstrated that connections to overcomplete independent component analysis…

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因果顺序的置信集合AI 解读

Causal discovery procedures aim to deduce causal relationships among variables in a multivariate dataset. While various methods have been proposed for estimating a single causal model or a single equivalence class of models, less attention has been given to quantifying uncertainty in causal discovery in terms of confidence statements. A primary challenge in causal discovery of directed acyclic graphs is determining a causal ordering among the variables, and our work offers a…

用不完全 U 统计量检验可能不规则模型中的大量约束AI 解读

Abstract We consider the problem of testing a null hypothesis defined by equality and inequality constraints on a statistical parameter. Testing such hypotheses can be challenging because the number of relevant constraints may be on the same order or even larger than the number of observed samples. Moreover, standard distributional approximations may be invalid due to irregularities in the null hypothesis. We propose a general testing methodology that aims to circumvent…

潜变量模型可识别性的半途径准则AI 解读

We consider linear structural equation models with latent variables and develop a criterion to certify whether the direct causal effects between the observable variables are identifiable based on the observed covariance matrix. Linear structural equation models assume that both observed and latent variables solve a linear equation system featuring stochastic noise terms. Each model corresponds to a directed graph whose edges represent the direct effects that appear as…

随机向量独立性的普遍一致、完全分布自由秩检验AI 解读

Rank correlations have found many innovative applications in the last decade. In particular, suitable rank correlations have been used for consistent tests of independence between pairs of random variables. Using ranks is especially appealing for continuous data as tests become distribution-free. However, the traditional concept of ranks relies on ordering data and is, thus, tied to univariate observations. As a result, it has long remained unclear how one may construct…

基于中心向外秩与符号的分布无关一致独立性检验AI 解读

This article investigates the problem of testing independence of two random vectors of general dimensions. For this, we give for the first time a distribution-free consistent test. Our approach combines distance covariance with the center-outward ranks and signs developed by Marc Hallin and collaborators. In technical terms, the proposed test is consistent and distribution-free in the family of multivariate distributions with nonvanishing (Lebesgue) probability densities.…

Kronecker 协方差极大似然估计的存在性与唯一性AI 解读

In matrix-valued datasets the sampled matrices often exhibit correlations among both their rows and their columns. A useful and parsimonious model of such dependence is the matrix normal model, in which the covariances among the elements of a random matrix are parameterized in terms of the Kronecker product of two covariance matrices, one representing row covariances and one representing column covariance. An appealing feature of such a matrix normal model is that the…

基于秩相关最大值的高维一致独立性检验AI 解读

Testing mutual independence for high-dimensional observations is a fundamental statistical challenge. Popular tests based on linear and simple rank correlations are known to be incapable of detecting nonlinear, nonmonotone relationships, calling for methods that can account for such dependences. To address this challenge, we propose a family of tests that are constructed using maxima of pairwise rank correlations that permit consistent assessment of pairwise independence.…

等方差假设下的因果发现AI 解读

Summary Prior work has shown that causal structure can be uniquely identified from observational data when these follow a structural equation model whose error terms have equal variance. We show that this fact is implied by an ordering among conditional variances. We demonstrate that ordering estimates of these variances yields a simple yet state-of-the-art method for causal structure learning that is readily extendable to high-dimensional problems.

非高斯条件下的高维因果发现AI 解读

Summary We consider graphical models based on a recursive system of linear structural equations. This implies that there is an ordering, $\sigma$, of the variables such that each observed variable $Y_v$ is a linear function of a variable-specific error term and the other observed variables $Y_u$ with $\sigma(u) < \sigma (v)$. The causal relationships, i.e., which other variables the linear functions depend on, can be described using a directed graph. It has previously been…

路径图的极大似然阈值AI 解读

Linear structural equation models postulate noisy linear relationships between variables of interest. Each model corresponds to a path diagram, which is a mixed graph with directed edges that encode the domains of the linear functions and bidirected edges that indicate possible correlations among noise terms. Using this graphical representation, we determine the maximum likelihood threshold, that is, the minimum sample size at which the likelihood function of a Gaussian…

含反馈环的结构方程模型中最大似然估计的计算AI 解读

Software for computation of maximum likelihood estimates in linear structural equation models typically employs general techniques from nonlinear optimization, such as quasi-Newton methods. In practice, careful tuning of initial values is often required to avoid convergence issues. As an alternative approach, we propose a block-coordinate descent method that cycles through the considered variables, updating only the parameters related to a given variable in each step. We…

基于秩相关之和的高维独立性检验AI 解读

We treat the problem of testing independence between $m$ continuous variables when $m$ can be larger than the available sample size $n$. We consider three types of test statistics that are constructed as sums or sums of squares of pairwise rank correlations. In the asymptotic regime where both $m$ and $n$ tend to infinity, a martingale central limit theorem is applied to show that the null distributions of these statistics converge to Gaussian limits, which are valid with no…

奇异模型的贝叶斯信息准则AI 解读

Summary We consider approximate Bayesian model choice for model selection problems that involve models whose Fisher information matrices may fail to be invertible along other competing submodels. Such singular models do not obey the regularity conditions underlying the derivation of Schwarz's Bayesian information criterion BIC and the penalty structure in BIC generally does not reflect the frequentist large sample behaviour of the marginal likelihood. Although large sample…

线性结构方程模型一般可识别性的半 trek 判据AI 解读

A linear structural equation model relates random variables of interest and corresponding Gaussian noise terms via a linear equation system. Each such model can be represented by a mixed graph in which directed edges encode the linear equations and bidirected edges indicate possible correlations among noise terms. We study parameter identifiability in these models, that is, we ask for conditions that ensure that the edge coefficients and correlations appearing in a linear…

《Wishart矩阵子式的矩》的更正AI 解读

Correction on Moments of minors of Wishart matrices by M. Drton and A. Goia (Ann. Statist. 36 (2008) 2261-2283), arXiv:math/0604488

线性结构方程模型的全局可识别性AI 解读

Structural equation models are multivariate statistical models that are defined by specifying noisy functional relationships among random variables. We consider the classical case of linear relationships and additive Gaussian noise terms. We give a necessary and sufficient condition for global identifiability of the model in terms of a mixed graph encoding the linear structural equations and the correlation structure of the error terms. Global identifiability is understood…

似然比检验与奇异点AI 解读

Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi-algebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions. We show that the assumptions of Chernoff’s theorem hold for semi-algebraic sets such that the asymptotics are determined by the…

Wishart 矩阵子式的矩AI 解读

For a random matrix following a Wishart distribution, we derive formulas for the expectation and the covariance matrix of compound matrices. The compound matrix of order m is populated by all m×m-minors of the Wishart matrix. Our results yield first and second moments of the minors of the sample covariance matrix for multivariate normal observations. This work is motivated by the fact that such minors arise in the expression of constraints on the covariance matrix in many…

边际独立性的二元变量模型AI 解读

Summary Log-linear models are a classical tool for the analysis of contingency tables. In particular, the subclass of graphical log-linear models provides a general framework for modelling conditional independences. However, with the exception of special structures, marginal independence hypotheses cannot be accommodated by these traditional models. Focusing on binary variables, we present a model class that provides a framework for modelling marginal independences in…