Suppose I have a confusion matrix $A$ for a set of points: entry $i,j$ is the fraction of time (over all $j$) when $i$ is present that $j$ is recognized (This means doubly stochastic. I don't think there are any other mathematical restrictions.).

This setup leads one to think of a subset that is has elements that are mutually more confusable within the subset than between pairs of elements where one is inside and the other outside the subset.

And vaguely this is reminiscent of a distance (or covariance) matrix. Except the confusion matrix is asymmetric but a distance/covariance matrix is not.

Is there a 'meaningful' (coherent, not totally crazy) mapping from a doubly stochastic matrix to a distance matrix? (where a distance matrix is symmetric, non-negative, obeys triangle inequality).

This is the application motivation for the question in Distance measure on weighted directed graphs. I think they are separable questions, but they can inform each other.