Is the Jaccard distance between probability vectors a metric?

Question

Let X and Y be probability vectors, meaning that X = $[x_1, x_2, ..., x_n]^T$, where $x_i\leq 1$ and $\sum_{i=1}^{n}x_i=1$ (Y is defined similarly).

Define the Jaccard distance as

\begin{equation} J_d = 1 - \frac{\textbf{X}\cdot\textbf{Y}}{\textbf{X}\cdot \textbf{X}+ \textbf{Y}\cdot\textbf{Y} - \textbf{X}\cdot\textbf{Y}} \end{equation}

Is $J_d$ a proper distance (i.e., metric)?

Answer 1

No, it is not. For fixed $Y$, fixed $Z=(z_1,\ldots,z_n)$ with $\sum z_i=0$, and small $t$ we have $J_d(Y\pm tZ, Y)\sim t^2 \frac{Z\cdot Z}{Y\cdot Y}$, but $$J_d(Y-tZ,Y+tZ)\sim 4t^2\frac{Z\cdot Z}{Y\cdot Y}>J_d(Y-tZ,Y)+J_d(Y,Y+tZ).$$