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A statistical critique of the Witztum et al paper
A.M. Hasofer
Posted February 18, 1998
Abstract. This paper examines whether the
significance test described in the paper Equidistant Letter Sequences in
the Book of Genesis by Witztum, Rips and Rosenberg was carried
out in accordance with accepted procedure, and whether the distance
metric used accomplishes its stated purpose. In the light of that
examination, it appears that the conclusion of the paper is unfounded.
Key words and phrases: Significance tests, alternative
hypoth esis, perturbation, Genesis, Equidistant letter
sequences.
Contents
 Introduction
 Significance
test procedure
 The
sample space and the null hypothesis
 The
alternative hypothesis
 A
critique of the "corrected" distance
 Conclusion
 Appendix A:
Details of the hypothesis testing procedure
 Appendix B:
Construction of the "corrected distance" counterexample
 References
The paper Equidistant Letter Sequences in the Book of Genesis
by Witztum, Rips and Rosenberg [7] is probably the Mathematical Statistics paper
which has enjoyed the widest dissemination ever, having been reproduced in
extenso in Drosnin's book [2] of which hundreds of thousands of copies have
been sold. Wild claims have been made about it. For example, Drosnin writes:"
The original experiment that proved the existence of the Bible code
(emphasis added) was published in a U.S. scholarly journal, Statistical
Science, a review journal of the Institute of Mathematical Statistics...In the
nearly three years since the RipsWitztumRosenberg paper was published, no one
has submitted a rebuttal to the math journal."(p.428). He also writes:"The three
referees at the math journal...started out skeptics and ended up believers". In
the remainder of this paper we shall refer to it as WRR. Equidistant letter
sequences will be referred to as ELS's.
The present paper examines whether the conclusion of the WRR paper is
justified.
Tests of significance are also known as tests of hypotheses because
they are based on setting up a "null hypothesis" and testing its significance
[1, p. 64].
The following steps must be followed when setting up a test of
hypothesis. Full details and references are given in Appendix A.
 Set up a sample space.
 Set up a null hypothesis.
 Set up a critical region.
 Set up the alternative hypothesis (or hypotheses).
 Investigate the power of the test.
We now examine to what extent WRR have carried out the accepted
procedure.
In previous work published on the ELS's (e.g. Michelson [6]) the
statistical calculations were based on the following hypothesis
H_{0}: each letter in the text is an independent discrete random
variable taking one of 22 values, each corresponding to a letter of the Hebrew
alphabet. The distribution of each discrete random variable is multinomial, with
the probability of each letter being equal to the relative frequency of the
letter in the whole text. Since the book of Genesis is 78,064 letters long, the
sample space contains 22^{78,064} texts, almost all of them gibberish.
In the final version of WRR, there are still some vestiges of the
earlier null hypothesis, e.g. in the calculation of the expected number of ELS's
for a word and the "expected distance" between two words (p.435). However, WRR
declare their null hypothesis H'_{0} to be as follows: They
define four overall measures of proximity P_{1}, P_{2},
P_{3} and P_{4} between the 32 personality date pairs
considered. For each of the 32!(2.63 x 10^{35}) permutations of the personalities they define the
corresponding statistic P_{1}. Thus the sample space is defined as the set of 32!
permutations of the personalities. The P_{1} are put in ascending order. The
null hypothesis is that P_{1} is just as likely to occupy any one
of the 32! places in this order as any other, and similarly for
P_{2}, P_{3} and P_{4} (p.431).
They provide the following motivation for the hypothesis: "If the phenomenon
under study were due to chance, it would be just as likely that (the correct
order of the names) occupies any one of the 32! places... as any other."
It must be emphasized at this point that this equiprobability
assumption has no frequency interpretation because we are faced here with a
unique object, namely the text of Genesis. The word "chance" used by the authors
is meaningless in this context. Because of this, the conclusion that "the
proximity of ELS's with related meanings in the book of Genesis is not due to
chance" is equally meaningless.
It must also be pointed out that equiprobability in a permutation
test usually follows by a symmetry argument from the assumption that the
underlying variables are independent and identically distributed [1, p. 182] as
the original null hypothesis assumed. But unfortunately here the symmetry
argument fails for the following reasons:
 Each personality may have several appellations, varying between
one and eight.
 Each date takes different forms, varying between one and
six.
 The lengths of the appellations and the dates vary between five
and eight.
 The sample of word pairs is constructed by taking each name of
each personality and pairing it with each designation of that personality's
date. Thus, when the personalities are permuted, the total number of word pairs
in the sample varies. We are not told in the paper of the range of variability.
For the identity permutation the sample consists of 298 words ([7, p. 436]), but
it appears that that number varies by more than 100 between different
permutations.([4]).
 5. The corrected distance is defined only when there are more than
nine perturbation triples for which a distance can be calculated. If not, the
corrected distance is not defined. We are not told how often this happens, or
what happens to the considered pair of words. Presumably, they are just dropped
from the sample. This would introduce an additional uncontrolled bias in the
calculation.
Thus the proposed null hypothesis H'_{0} does not have
any a priori justification even on the basis of the original hypothesis
H_{0}.
The problem of lack of frequency interpretation has been forcefully
highlighted by Matheron [5, p. 23] in the following words: "When we deal with a
unique phenomenon and a probabilistic model, that is a space (,, P) which is put in correspondence with this
unique reality,... (an anthropomorphic) illusion incites us to say that
everything happens, after all, as if the realized event had been 'drawn at
random' according to law P in the sample space .
But this is a misleadingly clear statement, and the underlying considerations
supporting it are particularly inadequate. What is the mechanism of this 'random
choice' that we evoke, which celestial croupier, which cosmic dice player is
shaking the iron dice box of necessity? This 'random draw' myth, for it is one,
(in the pejorative sense), is both useless and gratuitous. It is gratuitous, for
even if we assume that a unique random draw had been performed, once and for
all, and an element w_{0} had been selected, we would in
any case have no hope of ever reconstructing either the space or the probability P. For since we are
dealing with a unique event, the only source of information we possess is the
unique element w_{0}, which was chosen at first: all the rest,
the whole space and the
(almost) infinite ocean of possibilities it contained, will have disappeared,
erased forever by this unique draw. It is useless, that is, it has no
explanatory value, basically for the same reason: for the properties that we can
observe in our universe are contained in this unique element
w_{0}, and no longer depend on anything else. Thus we call
confidently ignore all the riches which slumbered in the other elements w, those
which have not been chosen. In any case, this element w_{0},
ours, had doubtlessly (an almost) zero probability of being drawn, and thus our
universe is 'almost impossible': nevertheless, it is the only universe given to
us, and the only one we can study."
Thus, the null hypothesis advanced by WRR must be considered as
"entirely hypothetical" ([1, p. 5])
In contrast with the situation with the null hypothesis, WRR
carefully avoid to make in the paper (or for that matter, apparently anywhere
else) any statement at all about the alternative hypothesis or hypotheses. This
in itself is a major flaw of the whole investigation, as is abundantly clear
from Appendix A, because in the absence of an alternative hypothesis it is
impossible to calculate the probability of a Type II error and there can
therefore be no grounds for rejecting the null hypothesis, no matter how
unlikely it might be in the light of the observations.
The WRR paper has however been widely publicized and those who have
made use of it have been far less reticent than the WRR authors. For example,
Drosnin [2] concludes: "We do have the scientific proof that some intelligence
outside our own does exist, or at least did exist at the time the Bible was
written." (p.50). In "public statements" attributed to two of the WRR authors
(E. Rips and D. Witztum) and circulated on the lnternet, commenting on Drosnin's
book, the above conclusion was not refuted and the expression "Bible codes" was
freely used.
It is therefore appropriate to consider as a possible alternative
that the book of Genesis was written by an intelligent being who could predict
the future and encoded this information in the text. It must of course be noted
that this hypothesis does not entail either that this being is benevolent or
that it is still in existence.
Some motivation for the research is given in the Introduction of WRR.
The approach is illustrated by the example of determining whether a text written
in a foreign language is meaningful or not. Pairs of conceptually related words
are considered and WRR write: "A strong tendency of such pairs to appear in
close proximity indicates that the text might be meaningful." They then declare
that the purpose of the research is "to test whether the ELS's in a given text
may contain 'hidden information'."
It is difficult to avoid the impression that the above reasoning is
just a naive anthropomorphism. On what grounds can we make any reasonable
assumptions abou the thought processes of a being that can predict the future?
Why should a text produced by such a being have properties similar to those of
an ordinary text in a foreign language written by a human?
But let us suppose that we agree to test the alternative hypothesis
that the encoder has actually put the appropriate dates nearest to each of the
names according to some distance measure P*. The alternative hypothesis
would then attribute a probability of one to the ordering where the correct
match had the smallest distance measure and zero to all others. The optimal
critical region would be just that one ordering, the size of the test would he
1/32! and its power wolrlrl be unity. Unfortunately, the data are not consistent
with this hypothesis, since none of the proposed measures falls in the proposed
critical region.
WRR estimale the rankings of their proposed measures P_{i
}, i = 1,...,4 by a MonteCarlo method. using one million permutations
chosen at random. This is perfectly acceptable, but what they do not explicitly
state is that even if we take the "best" of their measures of distance, namely
P_{4}, there are still an estimated 32! x 3/10^{6} 7.89 x 10^{29}
permutations
Book 
P_{1} 
P_{2} 
Genesis Exodus Leviticus Numbers Deuteronomy 
718 135,735 816,660 901,660 790,542 
2 193,315 947,387 920,919 759,428 
Table 1: Rank order of P_{1} and
P_{2} among one million P_{i.}
where at least two of the correct dates are not the
nearest ones to the appropriate names, but whose ranks are smaller than the rank
of the correct match! Some motivation for the encoder to choose such a bizarre
encoding must be provided by the authors, for otherwise we must conclude that
either their measures of distance are wildly off the mark, or else the
alternative hypothesis that there is encoded information about the 92
personalities in Genesis obeying their "close proximity" criterion is false. The
distance measure used by WRR will be examined more closely in the next Section.
Another problem is the fact that the experiment has been conducted
solely on the book of Genesis, while in all the public statements attributed to
Rips and Witztum reference is made to "codes in the Torah". But the word "Torah"
refers to all five books of the Pentateuch. In fact it is not known when the
Torah was divided into five books and by whom. There are also various opinions
as to where each book begins and ends. For example the Gaon of Vilna held that
Deuteronomy actually starts from the fifth verse of our present version ([6, p.
10]). The authors must explain why they chose to conduct the experiment solely
on Genesis. Moreover the argument provided must be an a priori one. This
is in view of the unfortunate fact that when conducted on the other four books
of the Pentateuch, their experiment (carried out on the second list of
personalities) failed to show any "significant" effect, at least as far as the
P_{1} and P_{2} statistics are concerned, as shown in
Table I ([4]). The results given for Genesis in the table are not exactly the
same as those appearing in the WRR paper because the experiment was carried out
with an updated version of the software, provided by WRR.
In WRR, a "corrected distance" between ELS's is defined on p.435. No
mathematical justification for the procedure is given. The "corrected distance"
between two words w and w' is supposed to be small when w is "unusually
close" to w', and 1 or almost 1 when w is "unusually far" from w'. There
are some problems in the exposition of the procedure:
 It will not work for skips of 1, 2 or ,3 because some of the
perturbed ELS's will not exist.
 The distance between two perturbed ELS's "is defined as the
distance between the ordinary (unperturbed) ELS's". This detinition is ambiguous
because the definition of distance between unperturbed ELS's involves f
and f', the distances between consecutive letters in the two ELS's.
But in the perturbed ELS's this distance is not fixed. In what follows we assume
that the f and f' of the unperturbed ELS's, but the actual minimal
distance between the letters of the perturbed ELS's are to be used.
 If the number of perturbed ELS's (out of 125) that actually appear
in the text is less than 10, the corrected distance is not defined. The paper
does not state what happens to the concerned pair. In the programs supplied by
WRR the pair is simply ignored (Source: McKay [4]).
But the really serious problem is that
 the word "usually" used in the justification is meaningless, since
we are dealing with a unique text,
 one can easily construct examples where the "corrected distance"
yields a result that is totally at variance with common sense. Such an example
is discussed further along in this Section. Full details are given in Appendix
B.
According to WRR, the "uncorrected proximity", , "very roughly measures the maximum
closeness of the more noteworthy appearances of w and w' as ELS's in
Genesis  the closer they are, the larger is " (p.435). It can be said in its favour
that when applied to pairs of words which occur only once and are all of the
same skip and length it reduces to a monotonically decreasing function of the
minimal distance along the text between the letters of the word pairs. This is
quite reasonable.
On the other hand, the "corrected distance" has nothing to do with
the actual position of the words whose distance is supposed to be measured. It
is based on the rank of the "uncorrected distance " between the two original words when
compared to the distances of pairs of "perturbed ELS's". But the sets of
"perturbed ELS's" are different for different pairs of words and so do not
provide any unified standard of comparison of distance.
For our counterexample we focus on just two pairs of fourletter
words, each having just one ELS, so as to keep the calculations simple and
transparent. They are denoted by w_{1}, w'_{1,}
w_{2 }and w'_{2}.. The minimum distance along the
text between the letters for the first pair is 90,000 letters and for the second
pair 30,000. The "corrected distance" for the first pair turns out to be zero
and for the second pair one.
Thus, according to the "corrected distance" w and w'_{1
}are "very close':, while w_{2 }and
w'_{2}. are "very distant". This is totally at variance with any
commonsense definition of distance between ELS's and contradicts all the
examples given in the Introduction of WRR to motivate the work. In addition, it
is clear from the details of the counterexample given in Appendix B that by
translating the set of perturbed ELS's without changing the position of the
original ELS's we can vary the "corrected distance" from 0 to 1. The only
limitations are due to boundary effects: if the two words are very close there
may not be enough space to insert perturbed ELS's. Conversely, if the two words
are near the beginning and the end of the text there may not be enough space to
translate the perturbed ELS's away from the two words.
That the phenomenon described in the counterexample is not artificial
can be illustrated by the data of the experiments themselves as follows: all
(matching) pairs of words from the two personality lists that did yield an and a c were collected
(Source: McKay [4]). There were 320 pairs. As an index of "corrected proximity"
we used 1  c . The range of was approximately (77  60,365). Since
we are interested in the ranking of proximities, a natural measure of the
concordance of the two indices is the rank correlation coefficient ([3. p.
494]). The overall value turns out to be 0.603. However, an examination of the
scattergram of the ranks indicates that the correlation is mainly due to small
and large values of .
Indeed, if we select 100 values of from the
middle of the range (specifically ranks 101 to 200, corresponding to 1450<
<3623) the rank
correlation between them and the corresponding 1  c's falls to 0.088. If
the pairs were a random sample the hypothesis of a zero rank correlation would
be accepted at the 20% significance level. The analysis thus supports the
conjecture that any association between and 1  c is not inherent but
entirely due to boundary effects.
The following flaws in the WRR paper have been documented:
 The probability distribution embodied in the null hypothesis is
purely hypothetical and cannot be justified on grounds of symmetry, as is
usually done for permutation tests.
 No alternative hypothesis is stated, so that the power of the
proposed test cannot be evaluated. A powerful reasonable alternative hypothesis,
based on the WRR heuristics, is proposed by the writer, but the experiment does
not reject the null hypothesis under that proposed alternative.
 No explanation is given for the choice of the particular book of
the Pentateuch on which the experiment was carried out.
 Some explanation must be given for the failure of the test to show
any significant effect on the other four out of five books of the
Pentateuch.
 The definition of "corrected distance", which is the basic
building block of the whole experiment, is shown not to achieve its purpose, for
it is easy to construct counterexamples where the "corrected distance" algorithm
leads to a result which is contrary to common sense. Moreover, analysis of the
correctly matched pairs of words in the two personality lists supports the
conjecture that any association between the proximity measure and the "corrected distance" c
is not inherent hut entirely due to boundary effects.
Until these flaws are remedied, the claims made in the paper must be
considered as statistically unfounded.
7 Appendix A: Details of the Hypothesis
Testing Procedure.
7.1 Setting up the null hypothesis.
The following quotation from Kendall and Stuart's Advanced Theory of
Statistics [3, p. 169] describes what type of null hypothesis is appropriate for
a significance test: "The kind of hypothesis which we test in statistics is more
restricted than the general scientific hypothesis. It is a scientific hypothesis
that every particle of matter in the universe attracts every other particle, or
that life exists on Mars; but there are not hypotheses such as arise for testing
from the statistical viewpoint. Statistical hypotheses concern the behavior of
observable random variables. More precisely, suppose that we have a set of
random variables x_{1},...,x_{n}. As before, we may
represent them as the coordinates of a point (x say) in the
ndimensional sample space, one of whose axes corresponds to each
variable. Since x is a random variable, it has a probability distribution, and
if we select any region, say w, in the sample space W, we may, (at least
in principle) calculate the probability that a sample point x falls in w say
. We shall say that any
hypothesis concerning is a
statistical hypothesis. In other words, any hypothesis concerning the behavior
of observable random variables is a statistical hypothesis."
7.2 Setting up the critical region.
The next step in the testing procedure is the setting up of the
critical region. We quote again from Kendall and Stuart [3, p. 171]:"To test any
hypothesis on the basis of a (random) sample of observations, we must divide the
sample space (i.e. all possible sets of observations) into two regions. If the
observed sample point x falls into one of these regions, say w, we shall reject
the hypothesis; if x falls into the complementary region, W  w, we shall accept
the hypothesis. w is known as the critical region of the test, and W  w
is called the acceptance region.
It is necessary to make it clear at the outset that the rather
peremptory terms 'reject' and 'accept' which we have used of a hypothesis under
test are now conventional usage, to which we shall adhere, and are not intended
to imply that any hypothesis is ever finally accepted or rejected in science. If
the reader cannot overcome his philosophical dislike of these admittedly
inapposite expressions, he will perhaps agree to regard them as code words,
'reject' standing for 'decide that the observations are unfavorable to' and
'accept' for the opposite. We are concerned to investigate procedures which make
such decisions with calculable probabilities of error, in a sense to be
explained.
Now if we know the probability distribution of the observations under
the hypothesis being tested, which we call call H_{0}, we can
determine w so that, given H_{0}, the probability of rejecting
H_{0 }(i.e. the probability that x falls in w) is equal to a
preassigned ,
i.e.
...The value is called the size of the test.."
7.3 Setting up the alternative hypothesis.
We continue the quotation: "Evidently, we can in general find many,
and often even an infinity, of subregions w of the sample space, all
obeying (1) Which of them should we prefer to the other? This is the problem of
the theory of testing hypotheses. To put it in everyday terms, which sets of
observations are we to regard as favoring, and which as disfavoring, a given
hypothesis?
Once the question is put in this way, we are directed to the heart of
the problem. For it is of no use whatever to know merely what properties a
critical region will have when H_{0} holds. What happens when
some other hypothesis holds? In other words, we cannot say whether a given body
of observations favors a given hypothesis unless we know to what alternative(s)
this hypothesis is compared. It is perfectly possible for a sample of
observations to be a rather 'unlikely' one if the original hypothesis were true;
but it may be much more 'unlikely' on another hypothesis. If the situation is
such that we are forced to choose one hypothesis or the other, we shall
obviously choose the first, notwithstanding the 'unlikeliness' of the
observations. The problem of testing a hypothesis is essentially one of choice
between it and some other or others. It follows immediately that whether or not
we accept the original hypothesis depends crucially upon the alternatives
against which it is being tested."
7.4 The power of a test.
We continue the quotation: "The (above) discussion... leads us
to the recognition that a critical region (or, synonymously, a test) must be
judged by its properties both when the hypothesis tested is true and when it is
false. Thus we may say that the errors made in testing a statistical hypothesis
are of two types:
 We may wrongly reject it, when it is true;
 We may wrongly accept it, when it is false.
These are known as Type I and Type II errors respectively. The
probability of a Type I error is equal to the size of the critical region used,
. The probability of a Type II
error is, of course, a function of the alternative hypothesis (say,
H_{1}) considered, and is usually denoted by . Thus
This complementary probability, 1  , is called the power of the test of
the hypothesis H_{0}, against the alternative hypothesis
H_{1}. The specification of H_{1 }in the last
sentence is essential, since power is a function of
H_{1}.
We seek a critical region w such that its power, defined at
(3), is as large as possible. Then, in addition to having controlled the
probability of Type I errors at , we shall have minimized the probability of a Type II error, . This is the fundamental idea,
first expressed explicitly by J. Neyman and E.S. Pearson, which underlies the
theory.
A critical region, whose power is no smaller than that of any other
region of the same size for testing a hypothesis H_{0} against
the alternative H_{1}, is called a best critical region
(abbreviated BCR) and a test based on a BCR is called a most powerful...
test."
8. Appendix B: Construction of the "corrected
distance" counterexample.
We construct a string of length 110,000 as follows. We use the first
22 letters of the latin alphabet (written in capitals) to represent the letters
of the Hebrew alphabet. We will focus on four "words"
w_{1}: ABCD, w'1: EFGH,
w_{ 2}: IJKL, and w'_{2}: MNOP. We use
the same notation as WRR for an ELS, namely (n, d, k) for the start, the
skip and the length.
We set:
w_{1} at (11992, 3, 4),w_{'1} at
(102001, 3, 4)
w_{2} at (41992, 3,4), w'_{2} at
(72001, 3, 4). These will be the only ELS's for these two words. We denote
the distance along the text between the last letter of w_{1} and
the first letter of w'_{1} by U_{1}. Here
U_{1}= 90000. Similarly we denote the distance along the text
between the last letter of w_{2} and the first letter of
w'_{2 }by U_{2}. Here U_{2}=
30000.
Pert. No. 
x 
y 
z 
1 2 3 4 5 6 7 8 9 
1 1 1 0 1 0 2 2 2 
0 0 1 1 1 1 0 0 2 
1 1 0 1 0 1 2 2 0 
Table 2: Perturbation triples.
Pert. No. 
w_{1} 
w'_{1} 
U_{1} 
w_{2} 
w'_{2} 
U_{2} 
1 2 3 4 5 6 7 8 9 
11962 11932 11902 11872 11842 11812 11782 11752 11722 
102031 102061 102091 102121 102151 102181 102211 102241 102271 
90060 90120 90180 90240 90300 90360 90421 90480 90540 
42022 42052 42082 42112 42142 42172 42202 42232 42262 
71971 71941 71911 71881 71851 71821 71791 71761 71731 
29940 29880 29820 29760 29700 29640 29580 29520 29460 
Table 3: Starting point of perturbed ELS's.
We now introduce 9 "perturbed" ELS's for each of the
four words, using the triples (x, y, z) given in Table 2:
As we are using a skip of 3, some of the perturbation triples used by
WRR will not work, namely those for which x+ y = 3 and/or x+ y+ z
= 3. We avoid them in our list of triples.
We note that, for all the triples we use, x+y+z = 0. Thus, the
perturbation only affects the position of the second and the third letter of the
words, but not the position of the first and last.
Table 3 gives the starting point of the perturbed ELS's for the four
words and the distances along the text..
The rest of the string is calculated arbitrarily with the 6 remaining
letters.
We now calculate the minimal distance L_{1} between a
letter of w_{l} and a letter of w'_{1}. We first
note that since the skip is 3, the only nonzero h_{i}'s, the
nearest integers to d/ i
i 
L_{1}(h_{i}) 
L_{2}(h_{i}) 
f(=f') 
1 2 3 4 
30000 45000 90000 90000 
10000 15000 30000 30000 
1
3 3 
Table 4: Distances between w_{1} and
w'_{1} for various h_{i}'s (1/2 rounded up), are h_{1} = 3,
h_{2} = 2, h_{3} = 1, h_{4} = 1.
The other h 's contribute zero to (w_{1},w'_{1}).
Here h'_{i}=h_{i}..
We note that the last letter of w_{ 1}, (D), has the rank
120011, so that for all nonzero h 's it is the first letter of the row.
Similarly, the first letter of w _{'1}, (E), has rank 102001, so that it
will also appear as the first letter of the row for all nonzero h 's.
The same considerations apply to w_{2} and w'_{2}, since H will
have rank 42001 and I will have rank 72001. The minimal cylindrical distances
between the letters of w_{1 }and w_{'1}, denoted by
L_{1}(h_{i}) and those between the letters of
w_{2} and w'_{2}, denoted by L_{2}(h_{i})
are given by Table 4.
It can be seen that when the skip is 3, if the distance U
along the text between the last letter of the first word w and the first letter
of the second word w' is a multiple of 6, (as is the case in our construction)
there are four finite cylindrical distances , namely U/3, U/2,
U and U.
Using the WRR definitions, we find for the proximity measures :(w, w') = 1.5556 x 10^{4 }and
sigma(w_{2},w'_{2})= 4.6667 x 10^{4}. This is as
it should be, since w _{2},w'_{2 }are much closer than w and
w_{1'}; by any reasonable criterion. Moreover, since in our case there
will be only one ELS (and one perturbed ELS for each perturbation triple) for
every word considered: we will have
^{(x,y,z)}(w, w'_{'})=^{(x,y,z)}(w,
w). 
(4) 
for every perturbation triple.
We now calculate the proximity measures of the perturbed ELS's. They
are given in 5. From Table 5 we conclude:
c(w_{1},w
'_{1})=0 
(5) 
c(w_{2},w
'_{2})=1 
(6) 
As pointed out in Section 5 this is totally
counterintuitive.
Pert. No. 
10^{4} x ^{(x,y,z)}
(w_{1},w'_{1}) 
10^{4} x ^{(x,y,z)}
(w_{2},w'_{2}) 
1 2 3 4 5 6 7 8 9 
1.5545 1.5535 1.5525 1.5514 1.5504 1.5494 1.5483 1.5473 1.5463 
4.6760 4.6854 4.6948 4.7043 4.7138 4.7233 4.7329 4.7425 4.7522 
Table 5: Proximity measures between perturbed
ELS's.
[1] Cox, D.R. and D. V. Hinkley. (1974) Theoretical
Statistics. Chapman and Hall.
[2] Drosnin, M. (1997) The Bible Code. Weidenfeld and Nicolson.
[3] Kendall, M.G. and A. Stuart. (1978) The Advanced Theory of
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[4] McKay, B. (1997) Private communication.
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