  It is common to hold prior beliefs that are not characterized by points in
the parameter space but instead are relational in nature and can be described
by a linear subspace. While some previous work has been done to account for
such prior beliefs, the focus has primarily been on point estimators within a
regression framework. We argue, however, that prior beliefs about parameters
ought to be encoded into the prior distribution rather than in the formation of
a point estimator. In this way, the prior beliefs help shape \textit{all}
inference. Through exponential tilting, we propose a fully generalizable method
of taking existing prior information from, e.g., a pilot study, and combining
it with additional prior beliefs represented by parameters lying on a linear
subspace. We provide computationally efficient algorithms for posterior
inference that, once inference is made using a non-tilted prior, does not
depend on the sample size. We illustrate our proposed approach on an
antihypertensive clinical trial dataset where we shrink towards a power law
dose-response relationship, and on monthly influenza and pneumonia data where
we shrink moving average lag parameters towards smoothness. Software to
implement the proposed approach is provided in the R package \verb+SUBSET+
available on GitHub.
