Compare the mean and covariance of a MCMC sample to a known value using a likelihood ratio test with a p-value.
For unit testing MCMC software. The returned p-values should be treated as a distance measure that has a distribution that is not too far away from the uniform, but the latter is not guaranteed. Simple testing can assert that p-values are bigger than a certain threshold, more complex testing can compare quantiles. See the docstrings of the exported functions mean_LR_pvalue and cov_LR_pvalue, and the unit tests.
Bai, Z., Jiang, D., Yao, J. F., & Zheng, S. (2009). Corrections to LRT on large-dimensional covariance matrix by RMT. The Annals of Statistics, 3822-3840.
Vats, D., Flegal, J. M., & Jones, G. L. (2015). Multivariate output analysis for Markov chain Monte Carlo. arXiv preprint arXiv:1512.07713.