  This paper studies constrained information projections on Banach spaces with
respect to a Gaussian reference measure. Specifically our interest lies in
characterizing projections of the reference measure, with respect to the
KL-divergence, onto sets of measures corresponding to changes in the mean (or
{\it shift measures}). As our main result, we give a portmanteau theorem that
characterizes the relationship among several different formulations of this
problem. In the general setting of Gaussian measures on a Banach space, we show
that this information projection problem is equivalent to minimization of a
certain Onsager-Machlup (OM) function with respect to an associated stochastic
process. We then construct several reformulations in the more specific setting
of classical Wiener space. First, we show that KL-weighted optimization over
shift measures can also be expressed in terms of an OM function for an
associated stochastic process that we are able to characterize. Next, we show
how to encode the feasible set of shift measures through an explicit functional
constraint by constructing an appropriate penalty function. Finally, we express
our information projection problem as a calculus of variations problem, which
suggests a solution procedure via the Euler-Lagrange equation. We work out the
details of these reformulations for several specific examples.
