# distance measure of two discrete probability histograms (distance between two vectors)

I have multiple sets of discrete probability histograms(vectors) and I want to measure the distance between each histogram. I have done some research but I am in doubt.

Literature suggest I could use the Bhattacharyya distance or the Hellinger distance (both are closely related). Which one should I use.

Some additional info. prob Histogram(vector) $A= (.18, .61, .16, .05)$ and prob Histogram (vector) $B= (.26, .55, .16, .03)$. I want to calculate the distance/difference among these two probability Histograms. The Bhattacharyya coefficient ($BC$) =

$$\sqrt{.18 \times .26}+\sqrt{.61 \times .55}+\sqrt{.16 \times .16}+\sqrt{.05 \times .03} \; .$$

Consequently the Bhattacharyya distance = $-\ln(BC)$ and Hellinger distance = $2 \sqrt{1-BC}$.

Is this the right measure and is this the right calculation?

• Is not the Hellinger Distance = $\sqrt {1−BC}$ ? That's what I have always used - and seems to be confirmed in a few references i have here. Not sure where the 2 factor comes into it? Commented Jan 7, 2022 at 11:53

  x <- rnorm(200)