I'm trying to understand, how the solid curve on the plot here was produced. The legend states that it is a binomial distribution, but as far as I know, this distribution produces integer numbers from 0 to infinity, not fractional from 0..1.

The following R code plots very similar curve

db<-dbinom(0:249, size=249, prob=0.5)
plot(x=seq(0, 1, along.with=db), y=db*9060003, type="l")

What confuses me, is the suspicion that I could adapt any binomial density to that one in the url above by choosing appropriate coefficients. Is it correct?

The original story is about comparison of two irises. A code is computed for each digital image of an iris, and these codes are compared. The iris code is a long bit string (2048 bits in the original work), and the comparison is the calculation of the Hamming distances between the bit strings (counting differing bits at the same positions).

Since some parts of an iris could be hidden with eyelids, eyelashes, etc, the masks are introduced, having 1 where code bits are valid, and 0 otherwise. This transforms the Hamming distance to the fractional Hamming distance:

  1. Counting differing bits, corresponding to ones in masks for both codes;
  2. This amount of valid differing bits is divided by the total number of bits that are valid on both codes.

What I don't fully understand, is how and why the binomial law describes the distribution of these fractional numbers.


1 Answer 1


I expect the drawing is actually a binomial ($X$) scaled by $n$, yielding the (discrete) proportion, $p=X/n$.

This is fairly common, but they should have made it explicit.

See this:

db <- dbinom(0:249, size=249, prob=0.5)
p <- (0:249)/249

enter image description here

If the probability of a bit being masked was constant for all bits, and masking of bits was independent (both conditions seem to be highly dubious), then the result would be binomial.

If the probability was varying but the masking was independent it would be distributed as Poisson-Binomial, but I'd expect adjacent bits to be highly correlated. It might be possible in some circumstances that the resulting distribution may be fairly reasonably approximated by a binomial, but I think it would make more sense just to go straight to normal approximation.

  • $\begingroup$ And is it OK that n differs from trial to trial? Because masks of valid bits are computed individually for each image. $\endgroup$
    – wl2776
    Jul 14, 2015 at 9:31
  • $\begingroup$ Ah; then there's no particular reason to expect the mean and variance to be constant across trials. We'd probably be looking at finite mixtures. $\endgroup$
    – Glen_b
    Jul 14, 2015 at 9:38
  • $\begingroup$ Thank you for response! Rereading the original paper (link to PDF file is given in the URL), I've found that author explicitly states "fractional binomial". The author's sample is very well described with Binomial distribution (fig 5 shows also QQ-plot). Could you suggest, how I can find the approximation of my sample, which is not so well described with Binomial? $\endgroup$
    – wl2776
    Jul 15, 2015 at 8:50
  • $\begingroup$ What does your sample look like? $\endgroup$
    – Glen_b
    Jul 15, 2015 at 11:07
  • $\begingroup$ They are skewed, like here: goo.gl/photos/M4pLKJnnaXXh8VbVA $\endgroup$
    – wl2776
    Jul 15, 2015 at 11:18

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