Title: Divergence and Sufficiency for Convex Optimization,
Abstract: Logarithmic score and information divergence appear in information theory,
statistics, statistical mechanics, and portfolio theory. We demonstrate that
all these topics involve some kind of optimization that leads directly to
regret functions and such regret functions are often given by a Bregman
divergence. If the regret function also fulfills a sufficiency condition it
must be proportional to information divergence. We will demonstrate that
sufficiency is equivalent to the apparently weaker notion of locality and it is
also equivalent to the apparently stronger notion of monotonicity. These
sufficiency conditions have quite different relevance in the different areas of
application, and often they are not fulfilled. Therefore sufficiency conditions
can be used to explain when results from one area can be transferred directly
to another and when one will experience differences.