< Back to Probability Distributions

I love a good mystery. I found a reference to the **Kullbach distribution** while researching unusual probability distributions. Of course, the first thing I did was to perform a Google search for “Kullbach distribution”. I got exactly one result from Haight [1]

So, what exactly is this mysterious distribution? A delve into the *Index to the Distributions of Mathematical Statistics* (IDMS) [1] provides a further clue:

The notation *GM* stands for geometric mean, and *Type III *is a skewed distribution similar to the binomial distribution [2], which does have the gamma function in the denominator.

The entry “*[2]251”* refers to page 251 of Kendall’s The Advanced Theory of Statistics , Volume 1 [3]:

But, page 251 of Kendall’s work does not have the “Kullbach distribution” listed, nor is there an entry for “Kullbach” in the back index.

The penny started to drop. This was a typo. A look at a later edition of the Index to the Distributions revealed that the spelling should be *Kullback distribution*.

## But, what *is *the “Kullbach Distribution?”

Solomon Kullback wrote about relative entropy in 1951 [4], where he described it as

“the mean information for discrimination between H

_{1}and H_{2}per observation from μ_{1}”

The asymmetric “directed divergence” (a distance metric) is now called the Kullback–Leibler divergence.

I did find reference to the formula listed in IDMS in Kullback’s 1931 paper *An Application of Characteristic Functions to the Distribution Problem of Statistics* [5] under a section titled “distribution of the geometric mean”, which is obtained from that of the arithmetic mean μ by the transformation of μ = log g^{n} .

Mystery solved!

**References**

[1] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[2] Abramowitz, M. and Stegun, I. A. (Eds.). (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover.

[3] KENDALL, M. G. The Advanced Theory of Statistics , Volume 1 , London: Charles Griffin and Co., 1945,

[4] Kullback, S.; Leibler, R.A. (1951). “On information and sufficiency”. *Annals of Mathematical Statistics*. **22** (1): 79–86. doi:10.1214/aoms/1177729694. JSTOR 2236703. MR 0039968.

[5] Kullback, S. (1934). An Application of Characteristic Functions to the Distribution Problem of Statistics. *The Annals of Mathematical Statistics*, *5*(4), 263–307. http://www.jstor.org/stable/2957612 p.277.

## One response to “Kullbach Distribution”

delve into the Index to the Distributions of Mathematical Statistics (IDMS) [1] provides a further clue

“Which?”