Model groups of variables and their interactions separately I want to decompose a multivariate pdf $p(x_1, x_2, x_3)$, where each $x_i$ is a collection of one or more random variables, into its "marginals" $p(x_1), p(x_2), p(x_3)$:
$p(x_1, x_2, x_3) = p(x_1)p(x_2)p(x_3) \cdot Z$
The goal is to be able to model interactions within the groups and between the groups separately, since I have different amounts of data for each group and only limited data that contains all groups for learning dependencies.
Intuitively I thought I could apply copula theory to determine an expression for the missing $Z$ that describes the dependencies between the groups of variables, but it seems that they are defined for univariate marginals.
Is there any extension of copula theory where the "margins" are groups of vairables, or some probability theory in general, to describe $Z$ apart from simply setting
$Z = \frac{p(x_1, x_2, x_3)}{p(x_1)p(x_2)p(x_3)}$ ?
Can $Z$ be expressed as some probability distribution over all variables?
 A: Short answer
In the example you stated above, your definition of $Z$ is the copula density function, $c$, of the copula function, $C$, which defines the dependence structure between the random variables $X_1$, $X_2$, and $X_3$.
Detailed answer
Sklar's theorem is often expressed 
$$C(F_1(x_1),F_2(x_2),\dots, F_n(x_n)) = H(x_1,x_2,\dots,x_n)$$, where $X_i$, are continuous random variables and $H$ is the joint distribution function of the random variables.  If we define $U_i = F_i(X_i)$, then we can rewrite Sklar's theorem as:
$$C(u_1, u_2, \dots, u_n) = H(F^{-1}_1(X_1), F^{-1}_2(X_2),\dots, F^{-1}_n(X_n))$$
Because $C$ is a proper distribution function in its own right, it has an associated density (called the copula density function), which we can derive by derivating along every dimension of the copula.  More specifically, we can write:
$$c(u_1,u_2,\dots,u_n) = \frac{\partial^n}{\partial u_1 \partial u_2 \cdots \partial u_n} C(u_1, u_2, \dots, u_n) = \frac{f(x_1, x_2, \dots, x_n)}{f(x_1)f(x_2)\cdots f(x_n)}$$
Where to go from here?
So, what you've laid out above in $Z$ is in fact exactly the copula density.  That being said, I'm not sure what you're trying to do, but some common things to do are to model your desired dependency based on a copula of choice (either data driven or expert knowledge driven).  Suppose you have some univariate random variables $X_1$, $X_2$, and $X_3$, and a desired dependence structure encapsulated by copula $C$.  Then, to generate samples of the desired multivariate distribution, you'd first sample $U_1$, $U_2$, and $U_3$ from $C$, and then apply $F^{-1}_i(u_i) = x_i$ to generate the samples from your joint distribution.  
Sources:


*

*Sklar's Theorem

*Generating samples from copulas
