This scheme might present an opponent
with the task of searching the cartesian product of many finite
independent nonuniform distributions.
If the real entropy is n bits less than the evident entropy then
a search order that estimates appropriate distributions for many of
the questions, may be 2n times faster.
Here is one way to use such estimates.
For each of the 20 or 30 less uniform distributions, compute
the probability of each combination.
Sort the resulting reals keeping the coresponding combination.
Try the most likely combinations first.
This scheme has several enhancements but it is hard to see that
they provide large improvements.