Suppose I have a zero-centered random variable $x\in \mathcal{D}^n$ where $\mathcal{D}$ is either $\{-1,1\}$ or $\mathbb{R}$, and its density has the following form in Einstein notation convention and symmetric tensor $M$
$$\log p(x) = x_i x_j x_k M_{ijk}$$
Is there a way to write higher order moments in terms of lower order moments, analogous to what Isserlis theorem does for the Gaussian?