  Many proper scoring rules such as the Brier and log scoring rules implicitly
reward a probability forecaster relative to a uniform baseline distribution.
Recent work has motivated weighted proper scoring rules, which have an
additional baseline parameter. To date two families of weighted proper scoring
rules have been introduced, the weighted power and pseudospherical scoring
families. These families are compatible with the log scoring rule: when the
baseline maximizes the log scoring rule over some set of distributions, the
baseline also maximizes the weighted power and pseudospherical scoring rules
over the same set. We characterize all weighted proper scoring families and
prove a general property: every proper scoring rule is compatible with some
weighted scoring family, and every weighted scoring family is compatible with
some proper scoring rule.
