Tropical Probability and Entropic Inequalities

Sherlock Holmes and doctor Watson take a ride on an air-balloon. After a sudden gust of wind takes them in an unknown direction, they spot a man on the ground and inquire about their location. After a short moment of consideration, the man answers: "You are on the air balloon". "This man is a mathematician!" -- concludes Sherlock, while wind carries the balloon further. "But how do you know?" -- wonders Dr. Watson. "For three reasons, dear Doctor. First, he thought before answering, second, his answer is absolutely correct and, finally, his answer is absolutely useless."

While we, with J. Portegies, were developing the theory of tropical probability spaces, the so fitting description of mathematical work given by Sherlock was rather frustrating, because the value of any theory is in its applications outside of itself.

Now that we have developed fairly sophisticated tools and are learning how to use them, the first fruits, though small and green, start appearing.

I will introduce the toolbox of tropical probability spaces and will show how it can be used to deduce a non-Shannon inequality for entropies of four random variables and their joints.