Equational reasoning is a central part of mathematics. The traditional theory links the equational logic and monads on the category of sets. More precisely, one can define an algebraic theory by giving a set of operations and equations. One can then show that the collection of algebras defined by these equations form the algebras for a monad on the category of sets. One has classical theorems like the completeness and variety theorems of Birkhoff. We consider a modified notion of equation where the equality symbol is annotated with a real number so we can write s =_e t with the interpretation that the terms "s" and "t" are within "e" of each other. We develop the metatheory and obtain analogues of Birkhoff's theorem. Furthermore, we show that this extended notion of equational definition yields algebras of monads on the category of metric spaces. What makes the theory interesting is the fact that well known metric spaces of probability distributions can be defined by such equations.

This is joint work with Radu Mardare and Gordon Plotkin.