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, integer-valued, compactly-supported, and positive 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.

翻译：数据变换对于参数回归模型的广泛应用至关重要。然而，在贝叶斯分析中，对变换和模型参数的联合推断通常依赖于限制性参数变换或计算效率低下、实现和理论分析繁琐的非参数表示，这限制了其在实际应用中的可用性。本文提出了一种简单、通用且高效的策略，用于对未知变换和所有回归模型参数进行联合后验推断。该方法通过将变换与自变量和因变量的边际分布联系起来，直接针对变换的后验分布，然后通过贝叶斯自助法部署贝叶斯非参数模型。关键的是，该方法实现了：(1) 在一般条件下（包括多种模型误设）的联合后验一致性，以及(2) 在重要特殊情形下对变换和所有参数的高效蒙特卡洛（而非马尔可夫链蒙特卡洛）推断。这些工具适用于多种数据领域，包括实值、整数值、紧支撑和正数据。仿真研究和实证应用证明了该策略在线性模型、分位数回归和高斯过程等半参数贝叶斯分析中的有效性和高效性。