Interpolating between the Jaccard distance and an analogue of the normalized information distance

摘要

Jiménez, Becerra and Gelbukh (2013) defined a family of‘symmetric Tversky ratio models’ S_(α,β), 0≤α ≤1,β >0. Each function D_(α,β) = 1? S_(α,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 andβ ≥ 1/(1? α). This result is formally verified in the Lean proof assistant. The extreme points of this parametrized space of metrics are V1 = D1/2,2 , the Jaccard distance and V∞ = D0,1 , an analogue of the normalized information distance of M. Li, Chen, X. Li, Ma and Vitányi (2004). As a second interpolation, in general, we also show that Vp is a metric, 1≤p≤∞ , where Δp(A, B)= (|B A|p+ |A B|p)1/p,Vp(A, B)=（Δp(A, B)）/（|A∩B|+ Δp(A, B) .）