How to compute P₂ ?

A formula is given in [WRR94, p.436] representing P₂ by a finite sum, but the sum is long (n terms). If you have "Maple V" or another program for mathematical computation, just ask it to compute the incomplete gamma function divided by the (complete) gamma function:

     GAMMA (n,s)   GAMMA (163, 251.33)              -8
P  = ----------- = ------------------- = 0.1096 * 10   ;
 2    GAMMA (n)      GAMMA (163)

here s = 251.33 is the sum of logarithms of (inverted) corrected distances, and n = 163 is the number of summands, for the second experiment of [WRR94]. (In fact, the default precision of 10 digits is not enough for my "Maple V" to get the correct answer for so large numbers; I did it with increased precision.) Note that (n-1)! = GAMMA (n) . If you have no such program, or want to get the result more quickly, I recommend you the following asymptotic formula:

          n-1  -s
         s    e             1
P    =   -------- * ---------------------  .
 2a       (n-1)!        n-1         1
                    1 - --- ( 1 - ----- )
                         s        s-n+3

Its relative error is less than 0.002 whenever the result is less than 10^-6. For examlpe, when s = 245 and n = 155 , the formula gives 0.2896*10^-9 while the exact value is 0.2898*10^-9. More examples (the first number is the approximate value, P_2a, the second one is the exact value, P₂):

A simple approximation for P₂

ln P2b = - 20.55 + L - 0.0115 L2

where

L = 17.19 - 0.365 ( s - 1.252 n ) .

The simple approximation makes it easy, to estimate the contribution of any personality (or appellation, or word pair) into the overall proximity measure P₂. For example, the personality considered on page 2 (Rabbi Avraham Av-Beit-Din of Narbonne) gives 12 word pairs, the corresponding sum of 12 logarithms being 4.828 + ... + 1.363 = 28.172 (as was said on page 2). Its contribution into L is determined by the value

s - 1.252 n = 28.172 - 1.252 * 12 = 13.15 .

Do you recognize the number? Look once again at page 1, the second list of outcomes. Yes, 13.15 appears there, it is just the last (maximal) element of the list. You may guess that each element of the list is s - 1.252 n for a personality, and you are right!

You may also guess that Rabbi Avraham Av-Beit-Din of Narbonne contributes more than any other personality to the overall proximity measure, and you are right, once again. Consider it in detail. The sum of the list (already calculated on page 2) is

s - 1.252 n = (-2.14) + (-2.00) + ... + 12.30 + 13.15 = 47.3 ,

which gives L = 17.19 - 0.365 * 47.3 = - 0.1 and ln P₂ = - 20.55 + L - 0.0115 L² = - 20.7 , that is, P₂ = exp (-20.7) = 0.11 * 10^-8 . It is just the overall proximity measure for the second experiment (calculated before as GAMMA(n,s)/GAMMA(n)).

What happens if we exclude the personality (Rabbi Avraham...)? Then s-1.252n decreases from 47.3 to 47.3 - 13.15 = 34.2, which gives L = 17.19 - 0.365 * 34.2 = 4.7 and ln P₂ = - 20.55 + L - 0.0115 L² = - 16.1 , that is, P₂ = exp (-16.1) = 0.10 * 10^-6 . The person contributes into P₂ the factor 0.011 . The factor can be roughly estimated via the value s-1.252n of the personality: exp ( - 0.365 * 13.15 ) = 0.0082 . Why not 0.011 ? Because of the nonlinear correction 0.0115 L² . The correction contributes the factor exp ( 0.0115 * ( 4.7² - 0.1² ) ) = 1.29 , and so, 0.0082 * 1.29 = 0.011 , as it should be.

After all, what is the relevance of the numbers on page 1 ?

Each number corresponds to a personality used by WRR. (Personalities that do not contribute are excluded.) The number is

s - 1.252 n ,

where (-s) is the logarithm of the product of all corrected distances for the personality, and n is the number of the distances. The relevant property of the linear form ( s - 1.252 n ) follows.

The close proximity of the sums (47.1 and 47.3) of the two lists of numbers (observed on page 1) is an equivalent reformulation of the close proximity (noted by Gil Kalai) of the two overall proximity measures.