Title: Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox,
Abstract: Statistical learning using imprecise probabilities is gaining more attention
because it presents an alternative strategy for reducing irreplicable findings
by freeing the user from the task of making up unwarranted high-resolution
assumptions. However, model updating as a mathematical operation is inherently
exact, hence updating imprecise models requires the user's judgment in choosing
among competing updating rules. These rules often lead to incompatible
inferences, and can exhibit unsettling phenomena like dilation, contraction and
sure loss, which cannot occur with the Bayes rule and precise probabilities. We
revisit a number of famous "paradoxes", including the three prisoners/Monty
Hall problem, revealing that a logical fallacy arises from a set of marginally
plausible yet jointly incommensurable assumptions when updating the underlying
imprecise model. We establish an equivalence between Simpson's paradox and an
implicit adoption of a pair of aggregation rules that induce sure loss. We also
explore behavioral discrepancies between the generalized Bayes rule, Dempster's
rule and the Geometric rule as alternative posterior updating rules for Choquet
capacities of order 2. We show that both the generalized Bayes rule and
Geometric rule are incapable of updating without prior information regardless
of how strong the information in our data is, and that Dempster's rule and the
Geometric rule can mathematically contradict each other with respect to
dilation and contraction. Our findings show that unsettling updates reflect a
collision between the rules' assumptions and the inexactness allowed by the
model itself, highlighting the invaluable role of judicious judgment in
handling low-resolution information, and the care we must take when applying
learning rules to update imprecise probabilities.