# Most Likely Transformations

arXiv: https://arxiv.org/abs/1508.06749 · HTML: https://mlsanity.com/paper/1508.06749/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Aug 2015.

## Built on by (most cited first, 20 of 20)

- [A review of predictive uncertainty estimation with machine learning](https://mlsanity.com/paper/2209.08307/) (2022)
- [Probabilistic Time Series Forecasts with Autoregressive Transformation Models](https://mlsanity.com/paper/2110.08248/) (2021)
- [Robust normalizing flows using Bernstein-type polynomials](https://mlsanity.com/paper/2102.03509/) (2021)
- [Short-Term Density Forecasting of Low-Voltage Load using Bernstein-Polynomial Normalizing Flows](https://mlsanity.com/paper/2204.13939/) (2022)
- [Deep interpretable ensembles](https://mlsanity.com/paper/2205.12729/) (2022)
- [What Makes Forest-Based Heterogeneous Treatment Effect Estimators Work?](https://mlsanity.com/paper/2206.10323/) (2022)
- [TranSUN: A Preemptive Paradigm to Eradicate Retransformation Bias Intrinsically from Regression Models in Recommender Systems](https://mlsanity.com/paper/2505.13881/) (2025)
- [Bernstein Flows for Flexible Posteriors in Variational Bayes](https://mlsanity.com/paper/2202.05650/) (2022)
- [Heterogeneous Treatment Effect Estimation for Observational Data using Model-based Forests](https://mlsanity.com/paper/2210.02836/) (2022)
- [Deep conditional transformation models for survival analysis](https://mlsanity.com/paper/2210.11366/) (2022)
- [Model-based causal feature selection for general response types](https://mlsanity.com/paper/2309.12833/) (2023)
- [Graphical Transformation Models](https://mlsanity.com/paper/2503.17845/) (2025)
- [An interpretable neural network-based non-proportional odds model for ordinal regression](https://mlsanity.com/paper/2303.17823/) (2023)
- [Let the Experts Speak: Improving Survival Prediction & Calibration via Mixture-of-Experts Heads](https://mlsanity.com/paper/2511.09567/) (2025)
- [Asymptotically Exact and Fast Gaussian Copula Models for Imputation of Mixed Data Types](https://mlsanity.com/paper/2102.02642/) (2021)
- [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)
- [Hybrid Bernstein Normalizing Flows for Flexible Multivariate Density Regression with Interpretable Marginals](https://mlsanity.com/paper/2505.14164/) (2025)
- [Conditional Rank-Rank Regression via Deep Conditional Transformation Models](https://mlsanity.com/paper/2603.07230/) (2026)
- [Scalable Learning of Multivariate Distributions via Coresets](https://mlsanity.com/paper/2603.19792/) (2026)

## Abstract (arXiv)

We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterisation of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set-up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterisation thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood-based estimation for conditional transformation models allowing the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.