  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.
