On model coefficient estimation using Markov chain Monte Carlo simulations: A potential problem and the solution

Markov chain Monte Carlo simulation is increasingly being considered as the tool of choice for model coefficient estimation. In almost all published papers, we use marginal posterior distributions for model coefficients to derive their point estimates, often the marginal means or medians. This note discusses a potential problem of using marginal posterior distribution for deriving point estimates. The problem arises when multiple model coefficients are correlated and the marginal distribution mean or median for each coefficient may not coincide with the respective coefficient value associated with the joint distribution mode. Furthermore, marginal distributions often overestimate model coefficients’ uncertainty. Consequently, we may obtain sub-optimal model coefficient estimates for subsequent inference. This note illustrates this problem through two examples and discusses a likely solution to the problem.