Is Partial-Dimension Convergence a Problem for Inferences from MCMC Algorithms?

Increasingly, political science researchers are turning to Markov chain Monte Carlo methods to solve inferential problems with complex models and problematic data. This is an enormously powerful set of tools based on replacing difficult or impossible analytical work with simulated empirical draws from the distributions of interest. Although practitioners are generally aware of the importance of convergence of the Markov chain, many are not fully aware of the difficulties in fully assessing convergence across multiple dimensions. In most applied circumstances, every parameter dimension must be converged for the others to converge. The usual culprit is slow mixing of the Markov chain and therefore slow convergence towards the target distribution. This work demonstrates the partial convergence problem for the two dominant algorithms and illustrates these issues with empirical examples.