Implementing MCMC: Multivariate estimation with confidence

This paper addresses the key challenge of estimating the asymptotic covariance associated with the Markov chain central limit theorem, which is essential for visualizing and terminating Markov Chain Monte Carlo (MCMC) simulations. We focus on summarizing batching, spectral, and initial sequence covariance estimation techniques. We emphasize practical recommendations for modern MCMC simulations, where positive correlation is common and leads to negatively biased covariance estimates. Our discussion is centered on computationally efficient methods that remain viable even when the number of iterations is large, offering insights into improving the reliability and accuracy of MCMC output in such scenarios.