  We introduce novel hypothesis testing methods for Gaussian graphical models,
whose foundation is an innovative algorithm that generates exchangeable copies
from these models. We utilize the exchangeable copies to formulate a
goodness-of-fit test, which is valid in both low and high-dimensional settings
and flexible in choosing the test statistic. This test exhibits superior power
performance, especially in scenarios where the true precision matrix violates
the null hypothesis with many small entries. Furthermore, we adapt the sampling
algorithm for constructing a new conditional randomization test for the
conditional independence between a response $Y$ and a vector of covariates $X$
given some other variables $Z$. Thanks to the model-X framework, this test does
not require any modeling assumption about $Y$ and can utilize test statistics
from advanced models. It also relaxes the assumptions of conditional
randomization tests by allowing the number of unknown parameters of the
distribution of $X$ to be much larger than the sample size. For both of our
testing procedures, we propose several test statistics and conduct
comprehensive simulation studies to demonstrate their superior performance in
controlling the Type-I error and achieving high power. The usefulness of our
methods is further demonstrated through three real-world applications.
