条件独立性是统计推断的一个基本概念。如何构造有效的条件独立性检验方法，是一项具有挑战性的工作。一方面，条件独立性检验统计量的构造涉及第三个随机向量的密度估计问题，在高维情况很难有好的表现。另一方面，在图像分析中，常常会遇到流形值的数据，比如fMRI分析中区分患者与正常人的大脑形状的差异。基于希尔伯特空间的条件独立性检验方法不能直接使用于这种类型的数据。因此，本项目将对条件独立作如下三个研究：1）针对高维的第三个随机向量，给出有效的条件独立性检验方法。2）在巴拿赫空间中定义一些只依赖于范数的条件依赖性量度，使得条件独立性和零量度具有等价性，给出量度的估计，作为条件独立性的检验统计量，并研究新统计量的渐进性质。3）针对各种数据类型和各种模型，研究新的条件独立性检验的实际应用性。

The concept of conditional independence is fundamental to statistical inference. It is a challenging work to construct a conditional independent test with high efficiency. On one hand, the existing tests of conditional independence suffer from circumstances when the third random vector is of high dimension. On the other hand, it is common to deal with manifold data in image analysis, for example to find the different shape of brain between normal person and patients in fMRI analyses. So this research project focuses on the following three aspects of conditional independence: 1)Find an efficient test of conditional independenc given the third random vector with high dimension; 2) Define a proper measure of conditional dependence in Banach space such that zero measure is equivalent to conditional independence, and then derive a new test of conditional independence, furthermore, study the asympotic properties of the test statistic. 3) Apply the new test for different data types and various models.