A Parametrized Family of Tversky Metrics Connecting the Jaccard Distance to an Analogue of the Normalized Information Distance

Jiménez, Becerra, and Gelbukh (2013) defined a family of “symmetric Tversky ratio models” Sα,β , 0 ≤ α ≤ 1, β > 0. Each function Dα,β = 1− Sα,β is a semimetric on the powerset of a given finite set. We show that Dα,β is a metric if and only if 0 ≤ α ≤ 12 and β ≥ 1/(1−α). This result is formally verified in the Lean proof assistant. The extreme points of this parametrized space of metrics are J1 = D1/2,2, the Jaccard distance, and J∞ = D0,1, an analogue of the normalized information distance of M. Li, Chen, X. Li, Ma, and Vitányi (2004).