A categorical characterization of relative entropy on standard Borel spaces

Recently there have been some exciting developments that bring categorical insights to probability theory and specifically to learning theory. These are reported in some recent papers by Clerc, Dahlqvist, Danos and Garnier.  There is also some recent work by Baez, Fritz and Leinster on a categorical characterization of relative entropy defined for distributions on finite sets. 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 called SbStat suitable for reasoning about statistical inference on standard Borel spaces. We define relative entropy as a functor into Lawvere’s category [0,∞] and we show convexity, lower semicontinuity and uniqueness. This is joint work with Nicolas Gagne.