  Bayesian Additive Regression Trees (BART) is a popular Bayesian
non-parametric regression model that is commonly used in causal inference and
beyond. Its strong predictive performance is supported by theoretical
guarantees that its posterior distribution concentrates around the true
regression function at optimal rates under various data generative settings and
for appropriate prior choices. In this paper, we show that the BART sampler
often converges slowly, confirming empirical observations by other researchers.
Assuming discrete covariates, we show that, while the BART posterior
concentrates on a set comprising all optimal tree structures (smallest bias and
complexity), the Markov chain's hitting time for this set increases with $n$
(training sample size), under several common data generative settings. As $n$
increases, the approximate BART posterior thus becomes increasingly different
from the exact posterior (for the same number of MCMC samples), contrasting
with earlier concentration results on the exact posterior. This contrast is
highlighted by our simulations showing worsening frequentist undercoverage for
approximate posterior intervals and a growing ratio between the MSE of the
approximate posterior and that obtainable by artificially improving convergence
via averaging multiple sampler chains. Finally, based on our theoretical
insights, possibilities are discussed to improve the BART sampler convergence
performance.
