Choice functions as a tool to model uncertainty

Our aim is to develop a tool for modelling different types of assessments about the uncertain value of some random variable. One well-know and widely used way to model uncertainty is using probability mass functions. However, such probability mass functions are not general enough to model, for instance, a total lack of knowledge. A very successful tool for modelling more general types of assessments is coherent sets of desirable gambles. These have many applications in credal networks, predictive inference, conservative reasoning, and so on. However, they are not capable of modelling beliefs corresponding to 'or' statements, for example the belief that a coin has two equal sides of unknown type: either twice heads or twice tails. Such more general assessments can be modelled with coherent choice functions. The first thing we do is relate coherent choice functions to coherent sets of desirable gambles, which yields an expression for the most conservative coherent choice function compatible with a coherent set of desirable gambles. Next, we study the order-theoretic properties of coherent choice functions. In order for our theory of choice functions to be successful, we need a good conditioning rule. We propose a very intuitive one, and show that it coincides with the usual one for coherent sets of desirable gambles, and therefore also leads to Bayes’s rule. To conclude, we show how to elegantly deal with assessments of indifference.