Non-Shannon-type Inequalities
Other, less trivial inequalities have been discovered among the entropies and joint entropies of four or more random variables, which cannot be derived from Shannon's basic inequalities. These are known as non-Shannon-type inequalities.
It was shown that
where the inclusions are proper for The last two sets are, in fact, convex cones.
Read more about this topic: Inequalities In Information Theory
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“In many places the road was in that condition called repaired, having just been whittled into the required semicylindrical form with the shovel and scraper, with all the softest inequalities in the middle, like a hog’s back with the bristles up.”
—Henry David Thoreau (1817–1862)