  Ordinal cumulative probability models (CPMs) -- also known as cumulative link
models -- such as the proportional odds regression model are typically used for
discrete ordered outcomes, but can accommodate both continuous and mixed
discrete/continuous outcomes since these are also ordered. Recent papers by Liu
et al. and Tian et al. describe ordinal CPMs in this setting using
non-parametric maximum likelihood estimation. We formulate a Bayesian CPM for
continuous or mixed outcome data. Bayesian CPMs inherit many of the benefits of
frequentist CPMs and have advantages with regard to interpretation,
flexibility, and exact inference (within simulation error) for parameters and
functions of parameters. We explore characteristics of the Bayesian CPM through
simulations and a case study using HIV biomarker data. In addition, we provide
the package 'bayesCPM' which implements Bayesian CPM models using the R
interface to the Stan probabilistic programing language. The Bayesian CPM for
continuous outcomes can be implemented with only minor modifications to the
prior specification and -- despite several limitations -- has generally good
statistical performance with moderate or large sample sizes.
