Is the marginal distribution of a maximum entropy distribution also a maximum entropy distribution?

Question

I am considering under what circumstances maximum entropy distributions are closed under marginalization. The main case I am interested is the following setting:

1. a finite set of discrete random variables $X_{1},\ldots,X_{n}$. To simplify notation, in the examples below I just use two random variables $X,Y$. The random variables are not independent.
2. a set of sufficient statistics $T_{1}, \ldots, T_{m}$ where $T_{i}(x_{1},\ldots,x_{n})$ returns a real number
3. a set of mean values $\mu_{1},\ldots,\mu_{m}$

I would like to start with the maximum entropy distribution $P^{\ast}(X,Y)$ that satisfies the moment matching constraints

$$E_{P}[T_{i}(X,Y)] = \mu_{i}.$$

It is well-known that this distribution can be represented in exponential form. Now consider the marginal sufficient statistics

$$T'_{i}(x) = \sum_{y} T_{i}(x,y).$$ with associated moment matching constraints $E_{Q}[T'_{i}(X)] = \mu_{i}$ and maximum entropy solution $Q^{\ast}$. Letting $P^{\ast}(x) = \sum_{y} P^{\ast}(x,y)$ be the marginal distribution for $P^{\ast}$, do we have that

$$P^{\ast}(x) = Q^{\ast}(x)? $$ If not true in general, are there conditions on the sufficient statistics that guarantee this condition?

Answer 1

Since $Q^*$ is of the form $$Q^*(x)\propto \exp\left\{\sum_j \lambda_j \mathbb{E}[T_j(x,Y)|X=x]\right\}$$ and $P^*$ is of the form $$P^*(x)\propto\sum_{y\in\mathcal Y} \exp\left\{\sum_j \zeta_j T_j(x,y)\right\}$$ and there is no reason this marginalisation property holds.