Augmentation Samplers for Multinomial Probit Bayesian Additive Regression Trees

The multinomial probit (MNP) (Imai and van Dyk, 2005) framework is based on a multivariate Gaussian latent structure, allowing for natural extensions to multilevel modeling. Unlike multinomial logistic models, MNP does not assume independent alternatives. Kindo et al. (2016) proposed multinomial probit BART (MPBART) to accommodate Bayesian additive regression trees (BART) formulation in MNP. The posterior sampling algorithms for MNP and MPBART are collapsed Gibbs samplers. Because the collapsing augmentation strategy yields a geometric rate of convergence no greater than that of a standard Gibbs sampling step, it is recommended whenever computationally feasible (Liu, 1994a; Imai and van Dyk, 2005). While this strategy necessitates simple sampling steps and a reasonably fast converging Markov chain, the complexity of the stochastic search for posterior trees may undermine its benefit. We address this problem by sampling posterior trees conditional on the constrained parameter space and compare our proposals to that of Kindo et al. (2016), who sample posterior trees based on an augmented parameter space. We also compare to the approach by Sparapani et al. (2021) that specified the multinomial model in terms of conditional probabilities. In terms of MCMC convergence and posterior predictive accuracy, our proposals are comparable to the conditional probability approach and outperform the augmented tree sampling approach. We also show that the theoretical mixing rates of our proposals are guaranteed to be no greater than the augmented tree sampling approach. Appendices and codes for simulations and demonstrations are available online.