  We introduce the concepts of Bayesian lens, characterizing the bidirectional
structure of exact Bayesian inference, and statistical game, formalizing the
optimization objectives of approximate inference problems. We prove that
Bayesian inversions compose according to the compositional lens pattern, and
exemplify statistical games with a number of classic statistical concepts, from
maximum likelihood estimation to generalized variational Bayesian methods. This
paper is the first in a series laying the foundations for a compositional
account of the theory of active inference, and we therefore pay particular
attention to statistical games with a free-energy objective.
