Augmentation Samplers for Multinomial Probit Bayesian Additive Regression Trees | JoVE Visualize

Area of Science:

Statistics
Machine Learning
Computational Statistics

Background:

The multinomial probit (MNP) framework, based on a multivariate Gaussian latent structure, offers advantages over multinomial logistic models by not assuming independent alternatives.
Bayesian additive regression trees (BART) have been integrated into MNP via multinomial probit BART (MPBART), utilizing collapsed Gibbs samplers for posterior sampling.
The efficiency of collapsed Gibbs samplers is contingent on simple sampling steps and fast Markov chain convergence, which can be challenged by the complexity of stochastic search for posterior trees.

Purpose of the Study:

To address the computational challenges in MPBART by proposing a novel posterior tree sampling strategy.
To compare the proposed method against existing MPBART approaches, including Kindo et al.'s (2016) augmented parameter space sampling and Sparapani et al.'s (2021) conditional probability specification.
To evaluate the proposed method's performance in terms of Markov chain Monte Carlo (MCMC) convergence and posterior predictive accuracy.

Main Methods:

The study proposes sampling posterior trees conditional on a constrained parameter space, contrasting with Kindo et al.'s (2016) method that uses an augmented parameter space.
A comparison is made with Sparapani et al.'s (2021) approach, which models the multinomial distribution using conditional probabilities.
The performance is assessed using MCMC convergence diagnostics and posterior predictive accuracy metrics.

Main Results:

The proposed conditional sampling approach demonstrates comparable MCMC convergence and posterior predictive accuracy to the conditional probability method.
The new method significantly outperforms the augmented tree sampling approach in both MCMC convergence and predictive accuracy.
Theoretical analysis confirms that the mixing rates of the proposed method are not inferior to the augmented tree sampling approach.

Conclusions:

The proposed method for sampling posterior trees in MPBART offers improved computational efficiency and predictive performance.
This approach provides a viable alternative to existing MPBART methods, particularly outperforming those relying on augmented parameter spaces.
The findings suggest that conditional sampling strategies can enhance the practical application of BART within the MNP framework.