  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 P\'olya-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.
