Tropical Probability Theory

There are several lines of research studying entropy-like quantities for a collection of random variables. Many constructions of such quantities are proposed, that satisfy certain conditions or have desired properties.

Tropical probability proposes to study entropy-like quantities in bulk. The closest analogy will be the relation between calculus and functional analysis.

Very roughly, entropy-like quantity associates a value to the collection of random variables. This association is required to be non-negative, additive (with respect to taking iid copies) and continuous with respect to some natural topology.

The word "tropical" refers to the certain "tropicalization" procedure, in spirit very similar to the "tropicalization" as used in algebraic geometry. Tropicalization is used for the construction of the "tropical cone" of diagrams of probability spaces -- a subcone in a certain Banach space and topology thereupon. The dual of the tropical cone is the cone of entropy-like quantities.

The ultimate goal would be to understand the space of all entropy-like quantities, from which one could then pick those needed for particular applications, such as causal inference, info decomposition, etc.

This area of research is little understood, open questions are behind every corner, and conjectures are abundant.

Date and time info
Wednesday, 11.00-12.30