Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss and Simpson’s Paradox

Abstract

Imprecise probabilities alleviate the need for high-resolution and unwarranted assumptions in statistical modeling. They present an alternative strategy to reduce irreplicable findings. However, updating imprecise models requires the user to choose among alternative updating rules. Competing rules can result in incompatible inferences, and exhibit dilation, contraction and sure loss, unsettling phenomena that cannot occur with precise probabilities and the regular Bayes rule. We revisit some famous statistical paradoxes and show that the logical fallacy stems from a set of marginally plausible yet jointly incommensurable model assumptions akin to the trio of phenomena above. Discrepancies between the generalized Bayes ($\mathfrak{B}$) rule, Dempster’s ($\mathfrak{D}$) rule and the Geometric ($\mathfrak{G}$) rule as competing updating rules for Choquet capacities of order 2 are discussed. We note that (1) $\mathfrak{B}$-rule cannot contract nor induce sure loss, but is the most prone to dilation due to “overfitting” in a certain sense; (2) in absence of prior information, both $\mathfrak{B}$- and $\mathfrak{G}$-rules are incapable to learn from data however informative they may be; (3) $\mathfrak{D}$- and $\mathfrak{G}$-rules can mathematically contradict each other by contracting while the other dilating. These findings highlight the invaluable role of judicious judgment in handling low-resolution information, and the care that needs to be take when applying updating rules to imprecise probability models.