I am reading this paper by Csiszar. Given a probability measure $R$ and a convex subset $\mathcal{E}$ of probability distributions, it defines ‘I-projection of R on $\mathcal{E}$’ (provided there exists a probability $P$ Such that the relative entropy of $P$ With respect to $R,$ that is $I(P||R),$ is finite) as a probability measure $Q$ that minimises the relative entropy wrt $R$ Over $\mathcal{E},$ that is $$I(Q||R)=\min\limits_{P\in \mathcal{E}}I(P||R).$$
A particular case is of interest to me where $X=\mathbb{R}^d\times\mathbb{R}^d.$ And the convex subset $\mathcal{E}\subseteq \mathcal{P}(X)$ is the set of probability measures defined as $$\mathcal{E}=\{\pi\in \mathcal{P}(\mathbb{R}^d\times\mathbb{R}^d): \pi_1=\mu, \int f_i(y)d\pi(x, y)=\int f_id\nu, i=1, \ldots,n \},$$ for some probability measures $\mu, \nu$ on $\mathbb{R}^d.$ And in my setting the measure $R=e^{-h(x, y)}dxdy$ For some function $h.$ It follows from the paper that the `I-projection of $R$ on $\mathcal{E}$’ has the following form $$\frac{\mu(x)e^{-h(x, y)}}{Z(x)}e^{\sum t_if_i(y)}\;dx\;dy,$$ where $t_i$ are suitably chosen constants.
I have the following two questions.
It is clear the teh first marginal of $Q$ Is actually $\mu.$ But what can we say about the second marginal? I have examples where it is not necessarily equal to $\nu.$ Does the second marginal belong to the exponential family?
Irrespective of whether we can compute the second marginal explicitly or not, I would like to understand if the second marginal of $Q,$ say $Q_2,$ is stochastic ally dominated by $\nu.$ That is, is it true that $$\int C(y)\;dQ(x, y)=\int C(y)\;dQ_2(y)\le \int C(y)d\nu(y),$$ for any convex function $C$?
