序数回归的延续比率Logit模型的结构化混合
Structured Mixture of Continuation-ratio Logits Models for Ordinal Regression
AI总结：
提出一种基于延续比率Logit混合的非参数贝叶斯序数回归方法，利用Pólya-Gamma数据增广实现高效后验模拟，避免了协变量效应的线性假设。
摘要：
我们开发了一种基于序数响应离散分布先验的非参数贝叶斯序数回归建模方法。先验概率模型由多项分布的结构化混合构建。我们利用延续比率Logit表示来构建混合核，并通过包含协变量线性函数的Logit Stick-Breaking过程定义混合权重。由此隐含的响应概率回归函数可表示为参数回归函数的加权和，且权重依赖于协变量。因此，该方法实现了灵活的序数回归关系，避免了协变量效应中的线性或可加性假设。通过先验概率模型的Kullback-Leibler支持度，我们正式探讨了模型的灵活性。模型的一个关键特征是，混合核和混合权重的参数均可与延续比率Logit回归结构相关联。因此，利用Pólya-Gamma数据增广，可设计出高效且易于实现的后验模拟方法。此外，模型基于类别特定参数的条件独立结构，通过部分并行采样获得额外的计算效率提升。除一般混合结构外，我们还研究了仅在混合核参数或仅在混合权重中引入协变量依赖的简化模型版本。对于所有提出的模型，我们讨论了先验设定方法，并开发了用于后验模拟的Markov链Monte Carlo方法。该方法通过几个合成和真实数据示例进行了说明。
英文摘要：
We develop a nonparametric Bayesian modeling approach to ordinal regression based on priors placed directly on the discrete distribution of the ordinal responses. The prior probability models are built from a structured mixture of multinomial distributions. We leverage a continuation-ratio logits representation to formulate the mixture kernel, with mixture weights defined through the logit stick-breaking process that incorporates the covariates through a linear function. The implied regression functions for the response probabilities can be expressed as weighted sums of parametric regression functions, with covariate-dependent weights. Thus, the modeling approach achieves flexible ordinal regression relationships, avoiding linearity or additivity assumptions in the covariate effects. Model flexibility is formally explored through the Kullback-Leibler support of the prior probability model. A key model feature is that the parameters for both the mixture kernel and the mixture weights can be associated with a continuation-ratio logits regression structure. Hence, an efficient and relatively easy to implement posterior simulation method can be designed, using Pólya-Gamma data augmentation. Moreover, the model is built from a conditional independence structure for category-specific parameters, which results in additional computational efficiency gains through partial parallel sampling. In addition to the general mixture structure, we study simplified model versions that incorporate covariate dependence only in the mixture kernel parameters or only in the mixture weights. For all proposed models, we discuss approaches to prior specification and develop Markov chain Monte Carlo methods for posterior simulation. The methodology is illustrated with several synthetic and real data examples.
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