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The Inflation Technique Completely Solves the Causal Compatibility Problem
Journal of Causal Inference ( IF 1.4 ) Pub Date : 2020-09-03 , DOI: 10.1515/jci-2018-0008
Miguel Navascués 1 , Elie Wolfe 2
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Abstract The causal compatibility question asks whether a given causal structure graph — possibly involving latent variables — constitutes a genuinely plausible causal explanation for a given probability distribution over the graph’s observed categorical variables. Algorithms predicated on merely necessary constraints for causal compatibility typically suffer from false negatives, i.e. they admit incompatible distributions as apparently compatible with the given graph. In 10.1515/jci-2017-0020, one of us introduced the inflation technique for formulating useful relaxations of the causal compatibility problem in terms of linear programming. In this work, we develop a formal hierarchy of such causal compatibility relaxations. We prove that inflation is asymptotically tight, i.e., that the hierarchy converges to a zero-error test for causal compatibility. In this sense, the inflation technique fulfills a longstanding desideratum in the field of causal inference. We quantify the rate of convergence by showing that any distribution which passes the nth-order inflation test must be On−1/2 $\begin{array}{} \displaystyle {O}{\left(n^{{{-}{1}}/{2}}\right)} \end{array}$-close in Euclidean norm to some distribution genuinely compatible with the given causal structure. Furthermore, we show that for many causal structures, the (unrelaxed) causal compatibility problem is faithfully formulated already by either the first or second order inflation test.

中文翻译:

通胀技术彻底解决因果相容问题

摘要 因果兼容性问题询问给定的因果结构图(可能涉及潜在变量)是否构成对图观察到的分类变量的给定概率分布的真正合理的因果解释。仅基于因果兼容性的必要约束的算法通常会遭受误报,即它们承认不兼容的分布与给定的图显然兼容。在 10.1515/jci-2017-0020 中,我们中的一个人介绍了膨胀技术,用于根据线性规划制定因果兼容性问题的有用松弛。在这项工作中,我们开发了这种因果兼容性松弛的正式层次结构。我们证明通货膨胀是渐近紧的,即,层次结构收敛到因果兼容性的零错误测试。从这个意义上说,膨胀技术满足了因果推断领域长期以来的需求。我们通过证明任何通过 n 阶膨胀测试的分布都必须是 On−1/2 $\begin{array}{} \displaystyle {O}{\left(n^{{{-} {1}}/{2}}\right)} \end{array}$-在欧几里得范数中接近某个与给定因果结构真正兼容的分布。此外,我们表明,对于许多因果结构,(非松弛的)因果兼容性问题已经通过一阶或二阶膨胀测试忠实地表达出来。我们通过证明任何通过 n 阶膨胀测试的分布都必须是 On−1/2 $\begin{array}{} \displaystyle {O}{\left(n^{{{-} {1}}/{2}}\right)} \end{array}$-在欧几里得范数中接近某个与给定因果结构真正兼容的分布。此外,我们表明,对于许多因果结构,(非松弛的)因果兼容性问题已经通过一阶或二阶膨胀测试忠实地表达出来。我们通过证明任何通过 n 阶膨胀测试的分布都必须是 On−1/2 $\begin{array}{} \displaystyle {O}{\left(n^{{{-} {1}}/{2}}\right)} \end{array}$-在欧几里得范数中接近某个与给定因果结构真正兼容的分布。此外,我们表明,对于许多因果结构,(非松弛的)因果兼容性问题已经通过一阶或二阶膨胀测试忠实地表达出来。
更新日期:2020-09-03
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