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Interpolating between the Jaccard distance and an analogue of the normalized information distance
Bjørn Kjos-Hanssen
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Jim\'enez, 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 D_(α,β) is a metric if and only if 0≤α≤1/2 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 𝒱₁=D_(1/2,2), the Jaccard distance, and 𝒱_∞=D_(0,1), an analogue of the normalized information distance of M. Li, Chen, X. Li, Ma, and Vit\'anyi (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)).
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