  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
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 some limitations, has generally good statistical performance with
moderate or large sample sizes.
