[2210.01516] Analysis of Conditional Randomization and Permutation Schemes by Application to Conditional Independence Tests

[Submitted on 4 Oct 2022]

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Overview: Examine the properties of two resampling scenarios, conditional randomization and conditional permutation schemes, related to conditional independence tests of discrete random variables $X$ and $Y$ given a random variable $Z$ . That is, we investigate the asymptotic behavior of the estimates of the probability vectors in such settings and establish an ordering between their asymptotic normality and the asymptotic covariance matrix. The results are used to derive the asymptotic distribution of the empirical conditional mutual information in these settings. Somewhat unexpectedly, the distributions of the two scenarios agree, despite the difference in the asymptotic distributions of the probability estimates. We also prove the validity of the permuted p-values ​​of the conditional permutation scheme. The above results justify consideration of resampled p-values ​​and a conditional independence test based on an asymptotic chi-square distribution with an adjusted number of degrees of freedom. Numerical studies show that tests based on asymptotic distributions adjusted for a limited number of permutations are a viable alternative to exact tests when the ratio of sample size to number of possible triple values ​​is greater than 0.5. Experiment shows. Both conditional permutation and conditional randomization scenarios. Furthermore, there is no significant difference in the performance of exact tests for conditional permutations and randomization schemes, the latter requiring knowledge of the conditional distribution of $X$ given $Z$ and the same conclusion being reached for both adaptive tests. applies to