Cumulative probability models (CPM), also known as cumulative link models, have been well-studied for ordinal and binary outcomes. Recently, Liu et al. (2017) showed that CPMs perform well for continuous outcomes with moderate to large sample sizes and are reasonably robust to link function misspecification, while McKinley et al. (2015) described the use of ordinal regression models in the Bayesian paradigm in the context of model choice. Combining these research threads, we examine the use of CPMs for continuous outcomes in a Bayesian framework by evaluating computational efficiency, robustness to model misspecification, and differences in software implementation. Because CPMs directly model the conditional distribution function they can handle any ordered outcome, including mixed distributions with discrete and continuous components, and are invariant to monotonic transformations. Quantiles and expectations are readily estimated from the conditional cumulative distribution while the use of Bayesian methodology allows interpretation of model parameters based on the posterior distribution.