Let $X$ is a discrete random vector of length three whose coordinates take integer values. We suppose approximate knowledge of all three 2-margins for $X$ and wish to create samples (say 10,000 examples) accordingly. Specifically we suppose that $I$, $J$ and $K$ are the dimensions of the lattice and for $i \in \{1,..,I\}$, $j \in \{1,..,J\}$ and $k \in \{1,..,K\}$ we desire:
$a_{j,k} \sim \frac{1}{I} \sum_{i=1}^I x_{i,j,k}$, $b_{i,k} \sim \frac{1}{J} \sum_{j=1}^J x_{i,j,k}$ and $c_{j,k} \sim \frac{1}{K} \sum_{k=1}^K x_{i,j,k}$
where $x_{i,j,k}$ denotes the number of occurrences of the value $(i,j,k)$ in our sample and $a$, $b$ and $c$ are known tables. Leaving ajar the precise joint distribution desired and the tolerance allowed in 2-margins, is there a fast way to sample $X$?
ps: I am a new user so could not add these tags: polytopes, contingency-tables, copulas, markov-bases,maximum-entropy