íåèöéå ïåøåã úàî

,Statistical Science -ä ìù '99 éàî ïåéìâá òéôåäì ãîåòä "Solving the Bible Code Puzzle" øîàîá

íéàáä íéøáãä úà (MBBK ,ïìäì) éòì÷ ìéâå ììä-øá äéî ,ïúð-øá øåøã ,é÷î ïãðøá íéøáçîä íéáúåë

:WRR íéø÷åçä åùîúùä åá éèñéèèñä ïçáîä úåãåà

"To correct the error in treating P_1-4 (that is, P₁, P₂, P₃ and P₄) as probabilities, Diaconis

proposed a method that involved permuting the columns of a 32X32 matrix, whose (i,j)th entry

was a single value representing some sort of aggregate distance between all the appellations of

rabbi i and all the dates of rabbi j. This proposal was apparently first made in a letter of May

1990 to the Academy member handling the paper, Robert Aumann, though a related proposal

had been made by Diaconis in 1988. The same design was again described by Diaconis in

September (Diaconis, 1990), and there appeared to be an agreement on the matter. However,

unnoticed by Diaconis, WRR performed the different permutation test described in Section 2."

(Section 3)

úîåòì ,"the test invented by WRR" :åðøîàîá íñøôúäù ïçáîä úà íéðëî íä ,10 ä÷ñéôá ,êùîäáå

."the permutation test of Diaconis"

ìéøôà-õøî) 27 'ñî "åàéìéìâ"á íñøôúäù ,àáä áúëîä úà é÷î ïãðøá ø"ã áúë ,ïëì íãå÷ ãåò

:('98

'îò 26 åàéìéìâ ,"úëøôåî äëøôä") ,úëøòîì íåèöéå ïåøåã ìù åáúëîá äøåîç úåòè ï÷úì éðåöøá

éãé ìò áúëåäù ïçáîá ùîúùä Statistical Science -á íñøôù éë"ðúä ïôåöä éåñéðù ïòåè àåä .(75

,éãéá íééåöî íäìù íé÷úòä øùà ,íéëîñî .øúåéá äòèîå úéø÷ù àéä åæ äòéá÷ .ñéðå÷àéã 'ôåøô

úåàîá äáåè äàöåú ïúð øùà ,øçà ïçáî êøòå ñéðå÷àéã ìù åðçáîî íìòúä àåäù íéçéëåî

íéáøåòîù íéøçà íéùðà éãé ìò øùåà øáë úàæ ìë .íéðù ùåìù åãéá åéä øáë íéðåúðä .íéðåî

ãéòî äî .ø÷ùä åúåà úà òéîùäì êéùîî íåèöéå øî ìáà ,øîàîìå ø÷çîì åôúåù íììëáå ,ïéðòá

."åúåðéîà ìò øáãä

ø"ã .åáù çöîä úåæòá òéúôäì êéøö ,(ùéàì éðééôåàä ïåðâéñá ,áâà ,çñåðîä) é÷î ø"ã ìù åáúëî

'ôåøô íò íëåñ ïëàù éåñéðä åäî äòéãé íåù ïéà åàéìéìâ éàøå÷ìù ,çéðäì ìåëé íðîåà é÷î

ú÷éåãîä äèèéöä ,äðäå .íðåøëéæá íééå÷ì åàéìéìâ éàøå÷ù çéðäì éàùø àåä ïéà êà .ñéðå÷àéã

:àéä åàìéìâá åîñøåôù éøáãî

ùîúùðù ñéðå÷àéã 'ôåøô òéöä ,äéðùä äîéùøä øåáòá íâ äãéãîä ìù äìåãâ äçìöä øçàì"

ìù úåòéúôîä úåàöåúäå ,åðéùò êë .äéðùä äîéùøä ìò äúåà ìéòôðå ,äùãç äãéãî úøåöá

"…éåñéðä

íøîàîá ,ììä-øá 'ôåøôå ïúð-øá ø"ã íò ãçé) åîöò é÷î ø"ã …áúëù íéøáãì ,äìà íéøáãì äååùð äúòå

:(53 'îò 25 'ñî åàìéìâá

íéëîñîä íéçéëåî úîàá äî øøáðå ,íéðééðòä êìäî úà àøå÷ä éðôì ùéâð äúò .ìëä äæ ïéà êà – ?íéäãî

.÷"ááî éãéáù

:Statistical Science -á åðøîàîá íñøôúäù éèñéèèñä ïçáîä ìò úîàä .á

PNAS -ì íåñøéôì ùâåä ,"Equidistant Letter Sequences in the Book of Genesis" ,åðìù øîàîä

øáçå íéìùåøéá úéøáòä äèéñøáéðåàá ä÷éèîúîì øåñôåøô àåäù ,ïîåà ìàøùé 'ôåøô éãé ìò

ééøôøä íò íéáúëî éôåìéçå úåçéù åì åéä ,äæ åðåéñð úøâñîá .íéòãîì úéà÷éøîàä äéîã÷àä

íäéðù åéä íä ,íéðåéãä íäéðéá åîëåñ åá áìùá '90 øáîèôñá .ñéðå÷àéã éñøô 'ôåøô

.ïî÷ìãë íéáúëî íäéðéá åôìçåäå ,ãøåôðàèñ úèéñøáéðåàá

:øáîèôñá 5 -á ïîåà 'ôåøôì àáä áúëîä úà çìù ñéðå÷àéã éñøô 'ôåøô

Professor Robert Aumann

Department of Economics

Mail Code 6072

Stanford University

Stanford, CA 94305

Dear Bob:

I am glad to report we are in agreement about the appropriate testing procedure for the paper

by Rips et al. A permutation test is to be performed. There are four basic sets of data/test statistics, I

will call them additive, multiplicative, with and without Rabbi. For each there is a 32X32 table of

distances. It is my understanding that for each such table, one million permutations will be performed.

For each permutation SIGMA_i=1³² t_ip(i) will be computed. This gives one million numbers/table.

Again for each the number SIGMA t_i will be located.

If it is within 1/4000 of the smallest table sums, that test is judged åeach the number

a success. If one of the four tests is successful, the whole experiment is.

In case of ties, the interval of ties will be broken at random. If half the proportion of such

breaks amount to better than 1/4000, that table is successful. Otherwise not.

I hope that the authors agree to make their findings public no matter what the outcomes. Please

let me know when you need from input from me.

Ð Sincerely,

Persi Diaconis

úçà éáâì "additive" øàåúá äðååëä äîì øåøá àì .1 :ìùîì .äæ áúëîá íéøåøá íðéà íéøáã äîë

òáøà ïðùéù òîùî .3 .äìáè ìë íéáéëøîä "distances" -ä íä äî øåøá àì .2 .úå÷éèñéèèñä ïî

äëéøö íéëéøàú/úåîù äîéùø åæéà åìéôà øåøá àì .4 .äðååëä äîì øåøá àìå ,úåðåù úåàìáè

.òöåîä éåñéðá ùîùì

:'90 øáîèôñá 7 -á ñéðå÷àéã 'ôåøôì àáä áúëîä úà ïîåà 'ôåøô áúë ,úàæ ìë øéäáäì éãë

Professor Persi Diaconis

Department of Statistics

Stanford University

Stanford, CA 94305

Dear Persi,

Thanks for your good letter of September 5, about the paper submitted by Rips et al. to the PNAS.

Since it's important to clarify the precise rules of a statistical test before performing it, allow me to set

down here a few points of clarification.

1. The same 1,000,000 permutations may be used for each of the four basic tests. The million will

consist of the identity permutation plus 999,999 others. All million will be different from each

other.

2. The sample to be examined is that of their "second experiment" (Table 3 of their submission).

For each of the four basic tests, the exact same procedures as reported on in their paper

(Tables 5 and 7) will be done for each of the 1,000,000 permutations. (Incidentally, "bunching"

or "twenty percent" might be a more suggestive name for the test you call "additive").

3. The precise tie-breaking rule (agreed on by phone today) is this: Out of the million

permutations, let there be s that are ranked smaller than the identity, and t with which it is tied

(excluding itself). Then the test is successful if and only if s+(t/2) < 250.

Again, with many many thanks for all your help on this,

Sincerely,

Bob Aumann

:åãé áúëá ïîåà 'ôåøô áúë ,áúëîä éìåùá

"given to Persi by hand in Sequoia hall, September 9, 1990, 2:50 PM. He looked it over and

approved."

.ñéðå÷àéã 'ôåøô éãé ìò åøùåà ïîåà 'ôåøô ìù åáúëîá íéèøôä ìë ,øîåìë

.åðéðôá äæ íåëéñ ïîåà 'ôåøô âéöä íéìùåøéì åàåáá

éðîéñ åáöåä úåàöåú íå÷îá øùàë ,íëåñîä éåñéðä éèøô éôì ÷åéãá ,ùãçî áúëð åðìù øîàîä

íéèôåù ãåòìå ñéðå÷àéã 'ôåøôì ïîåà 'ôåøô é"ò à"ðùúä úðù êìäîá çìùð ,äæ ùãç øîàî .äìàù

,øàåúîä éåñéðä ìò íúòã úååçì åù÷áúð íä .(íéòãîì úéà÷éøîàä äéîã÷àä éøáç íìåë)

.åúà íëåñîä ïî øçà éåñéð äæ éë äðòè äìòä àì ñéðå÷àéã 'ôåøô .äçìöäì óñ òåá÷ìå

íéèôåùä ïî íééðùå ñéðå÷àéã 'ôåøô úøæòá òá÷ ïîåà 'ôåøôù øçàì ,á"ðùúä óøåçá òöåá åîöò éåñéðä

éðîéñ íå÷îá øîàîá åöáåù úåàöåúä .úåéöåèåîøôä úøãñ úöøäì éèñéèèñä òøæä úà íéøçàä

.èåôéù ç"åã áåúëì åù÷áúð øùà ,íéèôåùì çååã äæ ìëå ,äìàùä

ñéðå÷àéã 'ôåøô éãéá åéä ,øîàîá èøåôîù éôë ,éåñéðä ìù ÷éåãîä øåàéúäù øáãä øåøá ïë íà

.éåñéðä òåöéá éðôì íéèôåùä øàùå

.'97 øàåðéá 17 íåéá ,øúåé øçåàî ììä-øá 'ôåøôì ïîåà 'ôåøô áúëù áúëî êåúî èèöà ,íåéñì

-å J (íúåà äðëî àåäù) íéáìùì òéâî àåäå ,åðìù ø÷çîä ìù äéâåìåðåøëä úà øàúî àåä øùàë

:áúåë àåä ,K

"J. The details of a formal test are agreed between Diaconis and Aumann (I'm trying to avoid

pronouns, because they often lead to confusion).

K. The formal test turns out significant at a level of 16 out of a million. (That is, the best result of

the four statistics is 4 out of a million, and then Bonferoni.)"

?ìéòì åâöåäù íéëîñîä úà íéøéëî MBBK íàä .â

íéëîñî ïîåà 'ôåøôî ììä-øá 'ôåøô äìáé÷ ('93 øáîèôñá 9) â"ðùúä ìåìà â"ë íåéá øáë !ïë

ïéðòá ñéðå÷àéã éñøô íò éìù úåáúëúää ìë [úà]" :(äååìðä áúëîá ïîåà 'ôåøô éøáãì) íéììåëä

".'úåùå ñôéø ìù äãåáòä

?øçù ïäì ïéàù úåðòè íéìòî úàæ íå÷îáå ,éèðååìøä òãéîä úà MBBK íéîéìòî òåãî ,ïë íà

ø"ã ìù åðåùìá ùîúùð íà ,íå÷î ìëî .íäìù úåøçàä úåðòèá ììë íéçåèá íðéà íäù äàøðë –

."íúåðéîà ìò øáãä ãéòî äî" :åàìéìâì ì"ðä åáúëîá é÷î