  We characterize a number of well known systems of approximate inference as
loss models: lax sections of 2-fibrations of statistical games, constructed by
attaching internally-defined loss functions to Bayesian lenses. Our examples
include the relative entropy, which constitutes a strict section, and whose
chain rule is formalized by the horizontal composition of the 2-fibration. In
order to capture this compositional structure, we first introduce the notion of
'copy-composition', alongside corresponding bicategories through which the
composition of copy-discard categories factorizes. These bicategories are a
variant of the Copara construction, and so we additionally introduce
coparameterized Bayesian lenses, proving that coparameterized Bayesian updates
compose optically, as in the non-coparameterized case.
