Augmentation Samplers for MultinomialProbit Bayesian Additive Regression Trees

Abstract

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