Title: The use of Proper Scoring Rules in Bayesian Model Selection
Abstract: A scoring rule S(x; q) provides a way of judging the quality of a quoted
probability density q for a random variable X in the light of its outcome
x. It is called proper if honesty is your best policy, i.e., when you believe
X has density p, your expected score is optimised by the choice q = p.
Every statistical decision problem induces a proper scoring rule, so there
is a very wide variety of these. Many of them have additional interesting
structure and properties. At a theoretical level, any proper scoring rule
can be used as a foundational basis for the theory of subjective probability.
At an applied level a proper scoring can be used to compare and improve
probability forecasts, and, in a parametric setting, as an alternative tool for
inference. We will discuss some characterisations and properties of proper
scoring rules, and we describe its use in the Bayesian model selection setting
in presence of improper priors.