  Posterior computation for high-dimensional data with many parameters can be
challenging. This article focuses on a new method for approximating posterior
distributions of a low- to moderate-dimensional parameter in the presence of a
high-dimensional or otherwise computationally challenging nuisance parameter.
The focus is on regression models and the key idea is to separate the
likelihood into two components through a rotation. One component involves only
the nuisance parameters, which can then be integrated out using a novel type of
Gaussian approximation. We provide theory on approximation accuracy that holds
for a broad class of forms of the nuisance component and priors. Applying our
method to simulated and real data sets shows that it can outperform
state-of-the-art posterior approximation approaches.
