Put trivariate PDF in terms of bivariate PDFs

Question

Let there be the random variables $X$, $Y$, and $Z$. Let all the bivariate PDFs $f_{X, Y}$, $f_{X, Z}$, and $f_{Y, Z}$ be known.

Can we write the unknown trivariate PDF $f_{X, Y, Z}$ in terms of the known bivariate PDFs?

Answer 1

Here is an example, obtained by tweaking a 2D counterexample: Let $(X,Y,Z)$ be such that $$ f_{X,Y,Z}(x,y,z)= \begin{cases} 2 \phi(x)\phi(y)\phi(z) & xyz>0\\ 0 & \text{otherwise} \end{cases} $$ where $\phi$ is the standard 1D Gaussian pdf. Then the bivariates $f_{X,Y}, f_{Y,Z}, f_{Z,X}$ are all standard 2D Gaussians, but of course $(X,Y,Z)$ is not Gaussian. Now both the standard 3D Gaussian and this $f$ give the same distribution of bivariates.

Answer 2

Let $f(x,y,z)=1$ when $x,y,z\in [0,1]$, and be zero otherwise. Then $f_{X,Y}$ is uniform on the unit square, and same for $f_{X,Z}$ and $f_{Y,Z}$. On the other hand, let $$ g(x,y,z)=1+\sin(2\pi (x+y+z)) \hspace{1cm}\text{for }x,y,z\in [0,1] $$ Then $g_{X,Y},g_{X,Z}$ and $g_{Y,Z}$ are also uniform on the unit square.