Let $p(x,y,z)$ be a probability distribution over 3 variables (suppose them discrete, but it shouldn't matter). I know that the distribution with maximal entropy which preserves the first order marginals $p(x),p(y),p(z)$ is simply the product $p(x)p(y)p(z)$.
What is instead the distribution with maximal entropy which preserves these marginals: $p(x,y),p(x,z),p(y,z)$? Or if there is no closed form, can I define it implicitly?
Thanks.