Suppose we have a linear regression problem. Let $F$ denote a set of input features, such that any subset of which is denoted $C \subseteq F$. The observed values for a particular subset of input features is denoted $\textbf{X}_{C}$. We also have a target $\textbf{y}$ and loss function $\ell$ (for which we get an in-sample estimate for its expected value).
The global Shapley value for the $d$-th feature in $1, \dots, |F|$ can be defined as: $$\psi_d = \sum_{C \subseteq F \setminus \{ d \}} w_{C} \left( v(C) - v(C \cup \{d\} \right) $$
Where $w$ is just a weight assigned to the marginal contribution denoted by $v(C) - v(C \cup \{d\})$, and $v$ is a characteristic function which represents the expected value of the loss for the coalition.
Now suppose we use Bayesian linear regression to get a posterior distribution of parameters. If we assume features are independant we can sample from the full joint posterior and generate the Shapley values to approximate the joint distribution of Shapley values. If my understanding is corrected, we are approximating this distribution: $$\Pr(\boldsymbol{\psi} | \textbf{y}) = \int \Pr(\theta | \textbf{y}) \prod_{d=1}^{|F|} \delta ( \psi_d - \psi_d(\theta) ) d \theta$$
where $\theta$ are the regression coefficients and $\Pr(\theta | \textbf{y})$ is the joint posterior. This works because we can effecitvely decompose the joint posterior because we have assumed independance.
But what if we dont assume independance? Then for each coalition we have to re-infer the new joint posterior. How would this integral look? Im struggling to figure this out.
Thanks.