# Bayesian Conditional Transformation Models

arXiv: https://arxiv.org/abs/2012.11016 · HTML: https://mlsanity.com/paper/2012.11016/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Dec 2020.

## Built on by (most cited first, 6 of 6)

- [Bernstein Flows for Flexible Posteriors in Variational Bayes](https://mlsanity.com/paper/2202.05650/) (2022)
- [Graphical Transformation Models](https://mlsanity.com/paper/2503.17845/) (2025)
- [Transformation Models for Flexible Posteriors in Variational Bayes](https://mlsanity.com/paper/2106.00528/) (2021)
- [Bayesian Discrete Conditional Transformation Models](https://mlsanity.com/paper/2205.08594/) (2022)
- [Monte Carlo inference for semiparametric Bayesian regression](https://mlsanity.com/paper/2306.05498/) (2023)
- [Scalable Learning of Multivariate Distributions via Coresets](https://mlsanity.com/paper/2603.19792/) (2026)

## Abstract (arXiv)

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.