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# Probability Filters as a Model of Belief; Comparisons to the Framework of Desirable Gambles

*Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications*, PMLR 147:42-50, 2021.

#### Abstract

We propose a model of uncertain belief. This models coherent beliefs by a filter, ${F}$, on the set of probabilities. That is, it is given by a collection of sets of probabilities which are closed under supersets and finite intersections. This can naturally capture your probabilistic judgements. When you think that it is more likely to be sunny than rainy, we have$\{ p | p(\textsc{Sunny}\xspace)>p(\textsc{Rainy}\xspace)\} \in {F}$. When you think that a gamble $g$ is desirable, we have $\{ p | \mathrm{Exp}_p[g]>0 \} \in {F}$. It naturally extends the model of credal sets; and we will show it captures all the expressive power of the desirable gambles model. It also captures the expressive power of sets of desirable gamble sets (with a mixing axiom, but no Archimadean axiom).