Accuracy, Estimates, and Representation Results

Abstract: Measures of accuracy usually score how accurate a specified credence depending on whether the proposition is true or false. A key requirement for such measures is strict propriety; that probabilities expect themselves to be most accurate. We discuss characterisation results for strictly proper measures of accuracy. By making some restrictive assumptions, we present the proof of the characterisation result of Schervish (1989) in an accessible way. We will also present the characterisation in terms of Bregman divergences and the relationship between the two characterisations. The new contribution of the paper is to show that the Schervish form characterises proper measures of accuracy for estimates of random variables more generally, by offering a converse to Schervish, Seidenfeld, and Kadane (2014, Lemma 1). We also provide a Bregman divergence characterisation in the estimates setting, using the close relationship between the two forms.