Frequency-calibrated belief functions: Review and new insights

Starting with Dempster's seminal work, several approaches to statistical inference based on belief functions have been proposed. Some of these approaches can be seen as implementing some form of prior-free Bayesian inference, while some others put the emphasis on long-run frequency properties and are more related to classical frequentist methods. This paper focusses on the latter class of techniques, which have been developed independently and had not been put in perspective until now. Existing definitions for frequency-calibrated belief functions as well as corresponding construction methods are reviewed, and some new notions and techniques are introduced. The connections with other frequentist notions such as confidence distributions and confidence curves are also explored. The different construction techniques are illustrated on simple inference problems, with a focus on interpretation and implementation issues.