# Monte Carlo inference for semiparametric Bayesian regression

arXiv: https://arxiv.org/abs/2306.05498 · HTML: https://mlsanity.com/paper/2306.05498/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 71 references that are in the map.
- First posted on arXiv 8 Jun 2023.

## Builds on (most cited first, 16 of 16)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [A general framework for Vecchia approximations of Gaussian processes](https://mlsanity.com/paper/1708.06302/) (2017)
- [Transforming Gaussian Processes With Normalizing Flows](https://mlsanity.com/paper/2011.01596/) (2020)
- [Compositionally-Warped Gaussian Processes](https://mlsanity.com/paper/1906.09665/) (2019)
- [Conjugate Bayes for probit regression via unified skew-normal distributions](https://mlsanity.com/paper/1802.09565/) (2018)
- [Practical bounds on the error of Bayesian posterior approximations: A nonasymptotic approach](https://mlsanity.com/paper/1809.09505/) (2018)
- [Bayesian Conditional Transformation Models](https://mlsanity.com/paper/2012.11016/) (2020)
- [A Class of Conjugate Priors for Multinomial Probit Models which Includes the Multivariate Normal One](https://mlsanity.com/paper/2007.06944/) (2020)
- [Bayesian Inference in Nonparanormal Graphical Models](https://mlsanity.com/paper/1806.04334/) (2018)
- [Simultaneous Transformation and Rounding (STAR) Models for Integer-Valued Data](https://mlsanity.com/paper/1906.11653/) (2019)
- [Bayesian Data Synthesis and the Utility-Risk Trade-Off for Mixed Epidemiological Data](https://mlsanity.com/paper/2102.08255/) (2021)
- [Learning non-Gaussian Time Series using the Box-Cox Gaussian Process](https://mlsanity.com/paper/1803.07102/) (2018)
- [Bayesian Inference for Regression Copulas](https://mlsanity.com/paper/1907.04529/) (2019)
- [Implicit Copulas from Bayesian Regularized Regression Smoothers](https://mlsanity.com/paper/1804.10397/) (2018)
- [Semiparametric discrete data regression with Monte Carlo inference and prediction](https://mlsanity.com/paper/2110.12316/) (2021)
- [Approximate Bayesian inference in semiparametric copula models](https://mlsanity.com/paper/1503.02912/) (2015)

## Abstract (arXiv)

Data transformations are essential for broad applicability of parametric regression models. However, for Bayesian analysis, joint inference of the transformation and model parameters typically involves restrictive parametric transformations or nonparametric representations that are computationally inefficient and cumbersome for implementation and theoretical analysis, which limits their usability in practice. This paper introduces a simple, general, and efficient strategy for joint posterior inference of an unknown transformation and all regression model parameters. The proposed approach directly targets the posterior distribution of the transformation by linking it with the marginal distributions of the independent and dependent variables, and then deploys a Bayesian nonparametric model via the Bayesian bootstrap. Crucially, this approach delivers (1) joint posterior consistency under general conditions, including multiple model misspecifications, and (2) efficient Monte Carlo (not Markov chain Monte Carlo) inference for the transformation and all parameters for important special cases. These tools apply across a variety of data domains, including real-valued, positive, and compactly-supported data. Simulation studies and an empirical application demonstrate the effectiveness and efficiency of this strategy for semiparametric Bayesian analysis with linear models, quantile regression, and Gaussian processes. The R package SeBR is available on CRAN.