Computation of Universal Objects for Distributions Over Co-Trees

For an ordered set, consider the model of distributions
P
for which an element that precedes another element is considered the more significant one in the sense that the implication
a
≤
b
⇒
P
(
a
) ≥
P
(
b
) holds. It will be shown that if the ordered set is a finite co-tree, then the universal predictor for the model or, equivalently, the corresponding universal code, can be determined exactly via an algorithm of low complexity. Natural relations to problems on the computation of capacity and on the determination of information projections are established. More surprisingly, a direct connection to a problem of isotone regression also appears possible.