TITLE: Projections in infinite dimensional statistical manifolds
QUESTION [6 upvotes]: I'm struggling to understand the geometry of projection for infinite dimensional statistical manifolds. In finite dimensions, a strictly convex smooth function $F$ defines a Bregman divergence. From this function/divergence, an finite-dimensional information-geometric structure is defined, which is dually-flat. The Riemannian metric is obtained from the Hessian of the strictly convex function, and the rest of the information is obtained from partial derivatives  (including the connection $\nabla^{F}$). In fact, there is a global coordinate system $[\theta]$ defined by the gradient $\nabla F$. In this case, it can be shown that unique projections from a point on the manifold to a submanifold exist, provided the submanifold is $\nabla$-flat (i.e. corresponds to a $\nabla$-affine subspace in the $[\theta]$-coordinate system). This, of course, broadly generalizes a projection to an affine subspace of Euclidean space. Does there exist a suitable analog for these facts in the infinite-dimensional information case?

REPLY [3 votes]: Are you trying to construct an infinite-dimensional (Hilbert) manifold of probability measures on a fixed measurable space? Or you simply want to find a Bergman divergence analog in order to generalize a divergence-like metric in an infinite-dimensional manifold? It is not entirely clear in the OP what kind of analog you are looking for.
In a finite-dimensional statistical manifold, for example an exponential family with natural parameterizations, the manifold does come with a canonical connection as you mentioned. In the finite-dimensional situation, since we can take a parameterization, the geometry of the statistical manifold is embedded in the parameter space. The image of such a projection to a $\nabla$-flat submanifold would correspond to a sub-family of the exponential family, also a restricted collection of parameters that define this sub-family.
In an infinite-dimensional statistical manifold, assuming that you want to compare two points; you can simply calculate the norms of their difference and other kind of similarity measures are possible [Harandi et.al.]. If you are talking about infinite-dimensional statistical manifold that consists of probability measures that cannot be The inner product structure like we have for low-dimensional parameter space, however, does not always exist. In a special case pointed out by [Newton2], we can use an $\alpha$-divergence (which is a generalization to Bergman divergence, see this post from stat.SE) to partially describe a similar inner product-like structure for an infinite-dimensional statistical manifold. The asymptotic results from geometric perspective [Kass&Vos] may generalize into infinite-dimensional situation if you can find a "good finite-dimensional basis approximation". However, a strict inner product structure is not always possible in infinite-dimensional situation, therefore projections may not generalize. If it does generalize, its meaning would likely be a sub-family of probability measures.
In another direction, if you wish to construct a realization (or draw samples) from an inifinite-dimensional statistical manifold, the starting point would be Dirichlet processes wiki. And [Newton] proposed to use transformation (Fenchel–Legendre transform) approach to construct new infinite-dimensional statistical manifolds in a more abstract way.
Reference
[Newton2] Infinite-dimensional statistical manifolds based on a balanced chart, 2016.
[Harandi et.al.] Bregman Divergences for Infinite Dimensional Covariance Matrices,2014.
[Newton] An infinite-dimensional statistical manifold modelled on Hilbert space,2012.
[Kass&Vos] Geometrical Foundations of Asymptotic Inference, 1997.