I'm approaching this question with the point of view that we consider the space $P$ of probability distributions on $\mathbb{R}^n$ (say with finite second moment), and consider the subspace $Q$ of those distributions with the additional requirement that all $n$ marginals are independent of one another. I.e. distributions in $Q$ can be expressed as a product, $$Q = \{ q\in P: q(\pmb{x}) = \prod_{i=1}^n q(x_i)\}.$$
I'm wondering about the geometric properties of $Q$. Is it dense in $P$? (Suppose $P$ is endowed with a distance metric, like the Wasserstein metric). Is it convex? We can talk about curves in $P$ since it is a Riemannian metric (Wasserstein space) -- can we do the same in $Q$?
Edit: there are some papers discussing this, but beyond my understanding. E.g.. They mention ``foliations" of the space $P$, which seems to give a hierarchy of models starting from the closest approximation $q \in Q$ to $p\in P$, and somehow getting more complex. Not sure how this helps.
Mean field optimization is always non-convex for any exponential family in which the state space $\mathcal{X}^m$ is finite. This can be seen very easily - the marginal polytope $\mathcal{M}(G)$ is a convex hull and $\mathcal{M}_F(G)$ contains all the extreme points of this polytope. This implies that $\mathcal{M}_F(G)$ is a strict subset of $\mathcal{M}(G)$ and is thus non-convex. For example, consider a two-node Ising model: $$\mathcal{M}_F(G) = \left\{ \tau_1, \tau_2 \in [0, 1] \quad \text{s.t.} \, \tau_{12} = \tau_1 \tau_2 \right\}$$
This has a parabolic cross section along $\tau_1 = \tau_2$ and hence it is non-convex.