Using coupling with the Gibbs sampler to assess convergence in animal models

The coupling method was investigated as a method to assess the convergence of the Gibbs sampler when drawing marginal inferences with an animal model. This method is based on the output of two Markov chains with different starting values but the same conditional deviates. The coupling method shows that the Gibbs sampler has an exponential convergence when the variance components are assumed to be known. All the variables in the model have the same rate of convergence, and it is closely related with the largest eigenvalue of a matrix derived from the coefficient matrix of the mixed model equations. Models including the variance components as unknowns showed a rate of convergence equal for all the variables, and this rate of convergence was approximately exponential. The coupling method provides an estimation of the convergence at the current iteration, and it does not require a post-Gibbs analysis as do the single chain-based methods. The coupling method is less computationally demanding than multiple chain methods, because only two chains are required to assess convergence.