  We give a conceptual treatment of the notion of joints, marginals, and
independence in the setting of categorical probability. This is achieved by
endowing the usual probability monads (like the Giry monad) with a monoidal and
an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If
the underlying monoidal category is cartesian monoidal, a bimonoidal structure
is given uniquely by a commutative strength. However, if the underlying
monoidal category is not cartesian monoidal, a strength is not enough to
guarantee all the desired properties of joints and marginals. A bimonoidal
structure is then the correct requirement for the more general case.
  We explain the theory and the operational interpretation, with the help of
the graphical calculus for monoidal categories. We give a definition of
stochastic independence based on the bimonoidal structure, compatible with the
intuition and with other approaches in the literature for cartesian monoidal
categories. We then show as an example that the Kantorovich monad on the
category of complete metric spaces is a bimonoidal monad for a non-cartesian
monoidal structure.
