Shapley effects for dependent inputs

Assume AC. On a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, let $d\ge 1$ be finite and let $X=(X_1,\dots ,X_d)$ be any jointly distributed finite family of random elements. Let $Y=f(X)\in L^2(\mathbb{P})$, and put $V=\mathrm{Var}(Y)$. For $S\subseteq [d]$, set ${\mathcal{G}}_S=\sigma (X_j:j\in S)$, with ${\mathcal{G}}_{\mathrm{\varnothing }}=\{\mathrm{\varnothing },\mathrm{\Omega }\}$, and $v(S)=\mathrm{Var}(\mathbb{E}[Y\mid {\mathcal{G}}_S])\mathrm{.}$ If $V>0$, let ${\mathfrak{S}}_d$ be the set of all $d!$ permutations of $[d]$, and let $P_i^{\pi }$ contain the indices occurring before $i$ in $\pi$. The Shapley effect of coordinate $i$ under this convention is ${\varphi }_i=\frac{1}{d!V}{\sum }_{\pi \in {\mathfrak{S}}_d}(v(P_i^{\pi }\cup \{i\})-v(P_i^{\pi }))\mathrm{.}$ If $V=0$, the normalized effects are undefined. This definition is for dependent inputs as well as independent ones; the independent Sobol ANOVA components are not used to define $v$.