A Measure of Local Sensitivity for Proper Scoring Rules in a Bayesian Setting

Suppose to express the uncertainty about an unobserved quantity $X \in \mathcal{X}$ by quoting a distribution $Q$ over $\mathcal{X}$, after which Nature reveals the value $x$ of $\mathcal{X}$. A {\em Scoring Rule} e $S(x, Q)$ provides a way of judging the quality of a quoted probability distribution $Q$ for in the light of its outcome $x$. It is called proper if honesty is your best policy, i.e. when you believe X has distribution P in M, your expected score is optimized by the choice Q=P. Every statistical decision problem induces a proper scoring rule. In this work we propose a general definition of local sensitivity index for Proper Scoring Rules from a Bayesian decision point of view. We show as this new index is an intrinsic characteristic of the class M