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Let $X,Y$ be two discrete random variables. Two joint mass distributions (couplings) with marginals $X$ and $Y$ and with entries $p_{i,j}=\mathbb{P}_1(X=i,Y=j)$ and $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ correspond to two matrices $(p_{i,j}), (p_{i,j}').$

Is there a linear transformation that maps $(p_{i,j})\mapsto(p_{i,j}')$?

If not, is there a way to move from a given coupling of $X,Y$ to any other coupling of $X,Y$?

The reason I ask is because I have a coupling of $X,Y$ and I wonder if a specific coupling exists. If there are any results that allow us to go from a given coupling to any other coupling, perhaps this can be used to provide the desired coupling or show that it doesn't exist.

The Substitute
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1 Answers1

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There is no linear transformation in general. Let $p_{i,j}=1/4$ for each $1\le i,j\le 2$ and $p'_{i,j}=1/2$ if $i=j$, 0 otherwise. Then we would need a matrix $A$ with $$\frac14 A \begin{bmatrix}1&1\\1&1\end{bmatrix}=\frac12\begin{bmatrix}1&0\\0&1\end{bmatrix}$$ But the rank of the product of two matrices is $\le$ the minimum of the ranks, so this is impossible.

Bjørn Kjos-Hanssen
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  • Do you know of any techniques to go from one coupling to another? – The Substitute Oct 08 '17 at 20:34
  • @TheSubstitute Actually assuming the distributions are not degenerate, i.e., the matrices have full rank, then you can do it with a linear transformation. Just multiply both sides by the inverse of a matrix to find $A$. – Bjørn Kjos-Hanssen Oct 10 '17 at 22:16