# Can a 3D joint distribution be reconstructed by 2D marginals?

Suppose we know p(x,y), p(x,z) and p(y,z), is it true that the joint distribution p(x,y,z) is identifiable? I.e., there is only one possible p(x,y,z) which has above marginals?

No. Perhaps the simplest counterexample concerns the distribution of three independent $\text{Bernoulli}(1/2)$ variables $X_i$, for which all eight possible outcomes from $(0,0,0)$ through $(1,1,1)$ are equally likely. This makes all four marginal distributions uniform on $\{(0,0),(0,1),(1,0),(1,1)\}$.
Consider the random variables $(Y_1,Y_2,Y_3)$ which are uniformly distributed on the set $\{(1,0,0),(0,1,0), (0,0,1),(1,1,1)\}$. These have the same marginals as $(X_1,X_2,X_3)$.