GEOMETRY OF THE FAITHFULNESS ASSUMPTION IN CAUSAL INFERENCE
GEOMETRY OF THE FAITHFULNESS ASSUMPTION IN CAUSAL INFERENCE
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DOI：
10.1214/12-aos1080
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发表时间：
2013-04-01
影响因子：
4.5
通讯作者：
Yu, Bin
中科院分区：
文献类型：
--
作者：
Uhler, Caroline;Raskutti, Garvesh;Yu, Bin
许多推断因果关系的算法严重依赖于忠实性假设。强加这一假设的主要理由是，不忠实分布的集合具有勒贝格测度零，因为它可以被视为超立方体中的超曲面的集合。然而，由于抽样误差的存在，仅凭忠实性条件是不足以进行统计估计的，人们已经提出了强忠实性条件，并假定它可以达到一致的或高维的一致性。与简单的忠诚度假设相反，不是强忠诚度的分布集具有非零的勒贝格度量，实际上，如我们在本文中所示，它可能会令人惊讶地大。我们从几何和组合的角度研究了强忠实条件，给出了各类有向无环图的强忠实分布的勒贝格测度的上下界。我们的结果暗示了PC算法的基本局限性，以及在高斯情况下基于部分相关测试的其他算法的潜在局限性。
Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a collection of hypersurfaces in a hypercube. However, due to sampling error the faithfulness condition alone is not sufficient for statistical estimation, and strong-faithfulness has been proposed and assumed to achieve uniform or high-dimensional consistency. In contrast to the plain faithfulness assumption, the set of distributions that is not strong-faithful has nonzero Lebesgue measure and in fact, can be surprisingly large as we show in this paper. We study the strong-faithfulness condition from a geometric and combinatorial point of view and give upper and lower bounds on the Lebesgue measure of strong-faithful distributions for various classes of directed acyclic graphs. Our results imply fundamental limitations for the PC-algorithm and potentially also for other algorithms based on partial correlation testing in the Gaussian case.