Put trivariate PDF in terms of bivariate PDFs 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?

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.
