I have a sequence $X^n$ of length $n$, where each $X_i$ takes a value from a finite set with probability vector $\mathbf{p} = [p_1, \ldots, p_K]^T$, i.e., $X_i \in [K]$, where $p_{X_i}(k) = p_k, k = 1, \ldots, K$. In this case, I can show that the Fisher Information matrix $I_{X^n}(\mathbf{p}) := \mathbb{E}[\nabla \log p(X^n)\nabla \log p(X^n)^T]= \text{diag}(\frac{n}{p_1}, \ldots, \frac{n}{p_K})$.
My question is: can we compute $I_{X^n}(\mathbf{p})$ when additional information is available regarding moments of the distribution $\mathbf{p}$ (e.g., mean and variance), i.e., $\mathbb{E}_\mathbf{p}[\phi_j(X)]= c_j, j = 1, \ldots, m$. My objective is to quantify the enhancement in the Cramer-Rao bound when incorporating side information about the moments. In fact, a lower bound on the Fisher Information would also work. While I am aware that there is existing work on constrained Cramer-Rao bounds, it tends to be intricate and occasionally unclear to me. I am hopeful though that one can establish more specific insights for this discrete setting with moment constraints (even if for some special cases such as mean and variance). Any assistance or guidance on this would be greatly appreciated.