I am given the projections of a multivariate probability distribution and asked to recover the multivariate distribution.
How do I derive the copula?
How do I use the copula to define the joint distribution function?
For continuous random variables $X,Y$ with probability density function $f_{X,Y}(x,y)$, the marginal probability distribution function of $X$ is defined as: $$ f_{X}(x) = \int_{\mathbb{R}} f_{X,Y}(x,y) \, dy$$ and similarly for $f_Y(y)$.
Here is a simple example diagram of what's going on. The black distribution is a projection and the red distribution is another projection of the multivariate distribution (in center):
In my case the probability density function $f_{X,Y}(x,y)$ lives in $(0,1)^3$ and has six projections onto each face of the boundary cube $[0,1]^2.$
$$ f_{X}(x)=\int_0^1 f_{X,Y}(x,y)~dy=K(s)\exp{\bigg(\frac{s}{\log(x)}\bigg)} $$
$$f_{Y}(y)=\int_0^1 f_{X,Y}(x,y)~dx=K(s)\exp{\bigg(\frac{s}{\log(y)}\bigg)}$$
The other four projections can be deduced based on symmetry, $s$ is the real parameter $s \in (0,\infty)$ and $K(s)$ normalizes (Modified Bessel function of the second kind).
Now this is all related to a distribution known as a Lorenz distribution. Clearly the marginals I just defined are univariate Lorenz distrubitions (real analytic ones). Based on Taguchi's proposed multivariate extension of the Lorenz curve I can "smell" the general shape of the distribution.
Here is Taguchi's proposed multivariate Lorenz distribution and what I need my distribution to "generally look like":
Although in my case the multivariate surface is smooth and maybe real analytic.