Let $(A,\varphi)$ be a unital noncommutative probability space over $\mathbb{C}$, so $A$ is a unital complex algebra with unit $1_A$ and $\varphi: A \to \mathbb{C}$ is a unital linear functional. Let $I$ be an index set, and for each $i \in I$ let $A_i \subset A$ be a unital subalgebra containing $1_A$. Assume the family $(A_i)_{i \in I}$ is free in the following explicit sense: for every integer $m \geq 1$, every tuple $(j_1,\dots,j_m) \in I^m$ satisfying $j_\ell \neq j_{\ell+1}$ for $1 \leq \ell \leq m-1$, and every choice of centered elements $x_\ell \in A_{j_\ell}$ satisfying $\varphi(x_\ell)=0$, one has

\begin{align*} \varphi(x_1x_2\cdots x_m)=0. \end{align*}

For every integer $n \geq 1$, every index tuple $(i_1,\dots,i_n) \in I^n$, and every choice of elements $a_k \in A_{i_k}$ for $1 \leq k \leq n$, the mixed moment

\begin{align*} \varphi(a_1a_2\cdots a_n) \end{align*}

is recursively determined by the index tuple $(i_1,\dots,i_n)$ and by the marginal data consisting, for each $i \in I$, of the algebra structure on $A_i$ and the restricted linear functional

\begin{align*} \varphi|_{A_i}: A_i \to \mathbb{C}. \end{align*}

Equivalently, the recursion may use only the given index pattern, multiplication inside a single component algebra $A_i$, and values of $\varphi$ on products formed entirely inside one $A_i$.