  The goal of this note is to provide a geometric setting in which generalized
arithmetic means are best predictors in an appropriate metric. This
characterization provides a geometric interpretation to the concept of
certainty equivalent. Besides that, in this geometric setting there also exists
the notion of conditional expectation as best predictor given prior
information. This leads to a notion of conditional preference and to the notion
of conditional certainty equivalent, which turns out to be consistent with the
notion of fair pricing.
