2
$\begingroup$

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

$\endgroup$
2
  • 1
    $\begingroup$ In the case of two dimensional contingency table, one could do a Gibbs sampler on a $2 \times 2$ block maintaining the row and column sum within that block only. Presumably that's still the case with 3-dimensional table with 2-dimensional marginal. The convergence rate however is far from clear. I heard that in the 2-dimensional case the best mixing time upper bound is $O(n^5)$. $\endgroup$
    – John Jiang
    Mar 10, 2011 at 0:04
  • $\begingroup$ Thanks John and yes, I share similar sentiments about the speed. $\endgroup$
    – Quant
    Mar 16, 2011 at 19:19

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.