  Penalized transformation models (PTMs) are a novel form of location-scale
regression. In PTMs, the shape of the response's conditional distribution is
estimated directly from the data, and structured additive predictors are placed
on its location and scale. The core of the model is a monotonically increasing
transformation function that relates the response distribution to a reference
distribution. The transformation function is equipped with a smoothness prior
that regularizes how much the estimated distribution diverges from the
reference distribution. These models can be seen as a bridge between
conditional transformation models and generalized additive models for location,
scale and shape. Markov chain Monte Carlo inference for PTMs can be conducted
with the No-U-Turn sampler and offers straightforward uncertainty
quantification for the conditional distribution as well as for the covariate
effects. A simulation study demonstrates the effectiveness of the approach. We
apply the model to data from the Fourth Dutch Growth Study and the Framingham
Heart Study. A full-featured implementation is available as a Python library.
