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

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Abstract
Jiménez, Becerra and Gelbukh (2013) defined a family of ‘symmetric Tversky ratio models’ |$S_{\alpha ,\beta }$|⁠, |$0\le \alpha \le 1$|⁠, |$\beta>0$|⁠. Each function |$D_{\alpha ,\beta }=1-S_{\alpha ,\beta }$| is a semimetric on the powerset of a given finite set. We show that |$D_{\alpha ,\beta }$| is a metric if and only if |$0\le \alpha \le \frac 12$| and |$\beta \ge 1/(1-\alpha )$|⁠. This result is formally verified in the Lean proof assistant. The extreme points of this parametrized space of metrics are |$\mathcal V_1=D_{1/2,2}$|⁠, the Jaccard distance and |$\mathcal V_{\infty }=D_{0,1}$|⁠, an analogue of the normalized information distance of M. Li, Chen, X. Li, Ma and Vitányi (2004).

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