## Permuted and Augmented Stick-Breaking Bayesian Multinomial Regression

*Quan Zhang, Mingyuan Zhou*; 18(204):1−33, 2018.

### Abstract

To model categorical response variables given their covariates,
we propose a permuted and augmented stick-breaking (paSB)
construction that one-to-one maps the observed categories to
randomly permuted latent sticks. This new construction
transforms multinomial regression into regression analysis of
stick-specific binary random variables that are mutually
independent given their covariate-dependent stick success
probabilities, which are parameterized by the regression
coefficients of their corresponding categories. The paSB
construction allows transforming an arbitrary cross-entropy-loss
binary classifier into a Bayesian multinomial one. Specifically,
we parameterize the negative logarithms of the stick failure
probabilities with a family of covariate-dependent softplus
functions to construct nonparametric Bayesian multinomial
softplus regression, and transform Bayesian support vector
machine (SVM) into Bayesian multinomial SVM. These Bayesian
multinomial regression models are not only capable of providing
probability estimates, quantifying uncertainty, increasing
robustness, and producing nonlinear classification decision
boundaries, but also amenable to posterior simulation. Example
results demonstrate their attractive properties and performance.

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