On topological and algebraic structures of categorical random variables
This work provides a theoretical framework for analyzing categorical random variables, which is incremental as it builds on existing entropy-based measures.
The authors tackled the problem of defining a metric for categorical random variables using entropy and symmetrical uncertainty, and they demonstrated that this metric can be extended to a quotient space with a compatible commutative monoid structure, where the monoid operation is continuous.
Based on entropy and symmetrical uncertainty (SU), we define a metric for categorical random variables and show that this metric can be promoted into an appropriate quotient space of categorical random variables. Moreover, we also show that there is a natural commutative monoid structure in the same quotient space, which is compatible with the topology induced by the metric, in the sense that the monoid operation is continuous.