6 research outputs found
A categorical characterization of relative entropy on standard Borel spaces
We give a categorical treatment, in the spirit of Baez and Fritz, of relative
entropy for probability distributions defined on standard Borel spaces. We
define a category suitable for reasoning about statistical inference on
standard Borel spaces. We define relative entropy as a functor into Lawvere's
category and we show convexity, lower semicontinuity and uniqueness.Comment: 16 page
A categorical characterization of relative entropy on standard Borel spaces
We give a categorical treatment, in the spirit of Baez and Fritz, of relative
entropy for probability distributions defined on standard Borel spaces. We
define a category suitable for reasoning about statistical inference on
standard Borel spaces. We define relative entropy as a functor into Lawvere's
category and we show convexity, lower semicontinuity and uniqueness
Postquantum Br\`{e}gman relative entropies and nonlinear resource theories
We introduce the family of postquantum Br\`{e}gman relative entropies, based
on nonlinear embeddings into reflexive Banach spaces (with examples given by
reflexive noncommutative Orlicz spaces over semi-finite W*-algebras,
nonassociative L spaces over semi-finite JBW-algebras, and noncommutative
L spaces over arbitrary W*-algebras). This allows us to define a class of
geometric categories for nonlinear postquantum inference theory (providing an
extension of Chencov's approach to foundations of statistical inference), with
constrained maximisations of Br\`{e}gman relative entropies as morphisms and
nonlinear images of closed convex sets as objects. Further generalisation to a
framework for nonlinear convex operational theories is developed using a larger
class of morphisms, determined by Br\`{e}gman nonexpansive operations (which
provide a well-behaved family of Mielnik's nonlinear transmitters). As an
application, we derive a range of nonlinear postquantum resource theories
determined in terms of this class of operations.Comment: v2: several corrections and improvements, including an extension to
the postquantum (generally) and JBW-algebraic (specifically) cases, a section
on nonlinear resource theories, and more informative paper's titl