置换和增广的断棒贝叶斯多项式回归
Permuted and Augmented Stick-Breaking Bayesian Multinomial Regression
摘要：
要建模的分类响应变量的协变量，我们提出了一个排列和增强棒打破（paSB）建设，一对一的映射观察到的类别随机排列的潜在棒。这种新的构造将多项式回归转换为对特定于棒的二元随机变量的回归分析，给定其协变量依赖的棒成功概率，这些随机变量是相互独立的，这些概率由其相应类别的回归系数参数化。paSB构造允许将任意交叉熵损失二进制分类器转换为贝叶斯多项式分类器。具体地说，我们参数化的负面的粘失效概率与一个家庭的协变量依赖的softplus功能，构建非参数贝叶斯多项式softplus回归，并转换贝叶斯支持向量机（SVM）到贝叶斯多项式SVM。这些贝叶斯多项式回归模型不仅能够提供概率估计，量化不确定性，增加鲁棒性，并产生非线性分类决策边界，而且还适合于后验模拟。实例结果证明了它们有吸引力的性质和性能。
英文摘要：
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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