  Recent developments in statistical regression methodology shift away from
pure mean regression towards distributional regression models. One important
strand thereof is that of conditional transformation models (CTMs). CTMs infer
the entire conditional distribution directly by applying a transformation
function to the response conditionally on a set of covariates towards a simple
log-concave reference distribution. Thereby, CTMs allow not only variance,
kurtosis or skewness but the complete conditional distribution to depend on the
explanatory variables. We propose a Bayesian notion of conditional
transformation models (BCTMs) focusing on exactly observed continuous
responses, but also incorporating extensions to randomly censored and discrete
responses. Rather than relying on Bernstein polynomials that have been
considered in likelihood-based CTMs, we implement a spline-based
parametrization for monotonic effects that are supplemented with smoothness
priors. Furthermore, we are able to benefit from the Bayesian paradigm via
easily obtainable credible intervals and other quantities without relying on
large sample approximations. A simulation study demonstrates the
competitiveness of our approach against its likelihood-based counterpart but
also Bayesian additive models of location, scale and shape and Bayesian
quantile regression. Two applications illustrate the versatility of BCTMs in
problems involving real world data, again including the comparison with various
types of competitors.
