# How does the Bhattacharyya distance doesn't satisfy triangle inequality?

Googling doesn't seem to show many informative results. I don't know if the concept is too trivial that I should know immediately or it's an old topic. It's either article / blogs repeating the wiki or just explaining the calculation.

In wiki, the Bhattacharyya distance:

$$D_B(p,q) = -\ln \left( BC(p,q) \right)$$

where the coefficient is given by

$$BC(p,q) = \sum_{x\in X} \sqrt{p(x) q(x)}$$

with these conditions

$$0 \leq BC ≤ 1 \space and \space 0 ≤ D_B ≤ ∞$$

From the wiki, it seems I should be able deduct that the distance doesn't satisfy the triangle inequality easily given the condition above. But I have no idea how.

The triangle inequality would be that $$D_B(p,q)\leq D_B(p,r)+D_B(r,q)$$ for all probability distributions $$p,q,r$$. So to show that the inequality does not hold, it is sufficient to find one counterexample.

One such counterexample is given by the following simple Bernoulli distributions: $$p=(0.1,0.9), \quad q=(0.9,0.1), \quad r=(0.5,0.5).$$ Then $$D_B(p,q) = -\ln(2\sqrt{0.09}) \approx 0.51$$ but $$D_B(p,r)=D_B(r,q)=-\ln(\sqrt{0.05}+\sqrt{0.45})\approx 0.11.$$

In general, I hack together a simple R script when searching for such counterexamples. (Or when I have a hunch and want to test it before thinking deeply about it. "Computers are cheap, and thinking hurts.") In the present case, a script like the following quickly points us in the right direction:

nn <- 2
normalize <- function(xx) xx/sum(xx)
DB <- function(pp,qq) -log(sum(sqrt(pp*qq)))

while ( TRUE ) {
pp <- normalize(runif(nn))
qq <- normalize(runif(nn))
rr <- normalize(runif(nn))
if ( DB(pp,rr) > DB(pp,qq)+DB(qq,rr) ) {
cat(pp,"\n",qq,"\n",rr,"\n")
break
}
}

• Now that I have a counter example convincing myself it's true. Is there other way to interpret such that I know it makes intuitive sense? I've seen someone seems to suggest if BC approaches 0, D_b will approach infinity, and somehow it is obvious the triangle inequality is not satisfied. Does this make any sense? Commented Sep 23, 2021 at 8:05
• Yes, that is precisely what is happening here. the probability masses of $p$ and $q$ are "very disjoint", so $BC(p,q)$ is very small, and $D_B(p,q)$ is very large. Taking the "detour across $r$" (detours is what the triangle inequality is all about) goes through a distribution $r$ whose mass is "much less disjoint" with either $p$ or $q$, so $BC(p,r)$ is much smaller, and $D_B(p,r)$ much larger. I don't know whether it gets much more intuitive than that... One could play around with $p=(t,1-t)$ and $q=(1-t,t)$ for various values of the parameter $t$. Commented Sep 23, 2021 at 8:14