Timeline for Generating random matrices with specific equality constraints
Current License: CC BY-SA 3.0
12 events
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| Aug 9, 2018 at 21:35 | history | edited | kjetil b halvorsen♦ |
edited tags
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| Nov 20, 2016 at 17:10 | comment | added | Xi'an | You are thus considering a distribution defined from an initial distribution (uniform) and a restriction to an implicit manifold. This recent answer is quite relevant. | |
| Nov 30, 2011 at 23:21 | comment | added | whuber♦ | Ah, then @Cardinal is (as usual) correct: the marginals will no longer be uniform and you're not requiring that they be so. In fact, they can be far from uniform. (But for medium to large $n$ they ought to be pretty close, because the constraints are so mild.) What you're asking is this: the constraints along with the bounds of the uniform distributions describe a polytope of $n^2-3$ dimensions; you would like to sample this polytope uniformly. | |
| Nov 30, 2011 at 23:21 | history | tweeted | twitter.com/#!/StackStats/status/142020499327815683 | ||
| Nov 30, 2011 at 20:40 | history | edited | Sami Liedes | CC BY-SA 3.0 |
Some further explanation of what I'm looking for (conditional probability)
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| Nov 30, 2011 at 20:20 | comment | added | Sami Liedes | What I'm thinking about is, if I'm not mistaken about the terminology, sampling the conditional probability distribution of uniformly distributed values conditioned with the given equations hold. Sure, the conditioned distribution is not necessarily uniform, but that's not a problem. Or in even other words, is there an efficient algorithm to replace the following: 1. Generate such a uniformly random matrix 2. If the equations do hold to within a difference of $\epsilon$, return the matrix; else goto 1 What I'm looking for is essentially this with $\lim \epsilon$ -> 0. | |
| Nov 30, 2011 at 20:09 | comment | added | whuber♦ | I'm not convinced of that incompatibility, @cardinal. Is there a proof? For instance, if we look at a similar set of constraints for 2 by 2 matrices (equal column sums, equal row sums, and total to unity) it turns out (trivially) that we can get all marginals to be uniform (in $[0,1/2]$, of course). But maybe this is a special case. | |
| Nov 30, 2011 at 17:52 | history | edited | cardinal | CC BY-SA 3.0 |
added 23 characters in body
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| Nov 30, 2011 at 17:51 | comment | added | cardinal | Related: stats.stackexchange.com/questions/17633/… | |
| Nov 30, 2011 at 17:43 | comment | added | cardinal | The Sinkhorn-Knopp algorithm is somewhat similar in that the goal there is to generate a matrix of nonnegative elements that satisfy individual row and column sum constraints. Quite a lot is known about its convergence properties. | |
| Nov 30, 2011 at 17:42 | comment | added | cardinal | Your two conditions (1) individual elements uniformly distributed and (2) your equality constraints are mutually incompatible. So, it's not entirely clear what you're aiming for. | |
| Nov 30, 2011 at 16:41 | history | asked | Sami Liedes | CC BY-SA 3.0 |