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I am wanting to sample from a distribution with uniform marginal distributions but with an interesting joint structure. Let the random vector be denoted $X = (X_1, X_2, \ldots, X_n)$ with $X_i \sim \text{uniform}(0,1)$. The random vector has the dependence structure $AX \leq b $ where A is a $q \times n$ matrix and $b$ a length $q$ vector (so I have $q$ linear constraints on $X$). The constraint I care about most is $\sum_{i=1}^N x_i \leq 1$.

If we consider a 2D example $X = (X_1, X_2)^T$ we have that $f_{1,2}(x_1, x_2) = f_1(x_1)f_2(x_2)c(x_1, x_2)$ hence the joint density is just the copula density (since uniform implies $F(x) = x$). By Sklar's theorem there must be a copula that exists. Most the copula I know of are essentially symmetric apart from e.g. the skewed Normal/Skewed-$t$ copula. Fairly naive random sampling doesn't give us the required density.

Any idea on how I could construct such a copula for direct sampling? Or even construct e.g. a markov chain which targets the correct distribution?

jcken
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    The duplicate unfortunately has an incorrect answer. But the question at https://stats.stackexchange.com/questions/159112 gives some nice pointers. – whuber Feb 04 '21 at 17:34
  • It seems like we can use the top answer here https://stats.stackexchange.com/questions/14059/generate-uniformly-distributed-weights-that-sum-to-unity but then if we drop $w_n$ (i.e. retain first $n-1$ elements of $w$) then that gives me my constraint! – jcken Feb 04 '21 at 17:48
  • That might work if the sole constraint is that the sum does not exceed unity. (But please note that your inequality is not the same as the equality constraint that defines the standard simplex. Notice especially that the marginals for a uniform distribution on the simplex are *not* uniform.) In the more general setting you propose, the problem is much more difficult, especially as $n$ grows. Do you need uniform marginals? How exactly does your dependence structure work--is it a condition applied to iid uniform variates or, *after applying the condition,* must the marginals be uniform? – whuber Feb 04 '21 at 18:13
  • Essentially I want inputs for a black-box computer simulation. The inputs sum to at most $1$ and are all positive. They don't need to be strictly uniform but I want good coverage of $[0,1]$ for each input which is why I thought a copula might be a good idea. – jcken Feb 04 '21 at 18:24
  • Unfortunately, those kinds of constraints are usually incompatible with uniform marginals *and* uniform sampling of the entire space. The uniform marginals push too much probability into the corners of the polytope. Thus, it would be a good idea to think a little more precisely about what you really need to accomplish. – whuber Feb 04 '21 at 18:28

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