Indifference, symmetry, and conditioning

In this talk about foundations, I intend to discuss the important consequences of introducing a notion of indifference in the field of inference and decision making under uncertainty, in the very general context of coherent sets of desirable options.

I’ll begin with a short discussion of the basic model—coherent sets of desirable options—and intend to show how it can capture many relevant aspects of so-called conservative probabilistic inference. In particular, I’ll very briefly recall how this model leads to coherent lower and upper previsions or expectations, conditioning, and coherent conditional (precise) previsions.

I’ll then discuss how a notion of indifference can be introduced into this context, and what its effects are on desirability models, through representation results. In this context, the conceptually different notions of conservative inference under indifference and updating under indifference make their appearance.

I’ll then conclude by presenting examples: specific useful and concrete instances of the abstract notions of option space and indifference. Such as:

• observing an event and conditioning a coherent set of desirable gambles on this observation;
• observing the outcome of a measurement and Lüders’ conditionalisation in quantum mechanics;
• exchangeability and de Finetti’s representation theorem in precise and imprecise probabilities contexts.