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

Jim\'enez, 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 $0\le\alpha \le \frac12$ 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 $J_1=D_{1/2,2}$, the Jaccard distance, and $J_{\infty}=D_{0,1}$, an analogue of the normalized information distance of M.~Li, Chen, X.~Li, Ma, and Vit\'anyi (2004).