Discussion of: Statistical analysis of an archaeological find

The Annals of Applie d Statistics 2008, V ol. 2, No. 1, 84–90 DOI: 10.1214 /08-A OAS99G Main articl e DO I: 10.1214/ 08-AOAS99 c  Institute of Mathematical Statistics , 2 008 DISCUSS ION OF: S T A TISTICAL A NA L YSIS OF AN AR C HAEOLOGICAL FIND By Randall Ingermanso n Ingermanson Communic ations , Inc. W e critique the analysis by A . F euerverger of an arc haeologi- cal find that h as b een alleged by some to b e t h e tom b of Jesus of Nazareth. W e show that his analysis rests on six fault y assumptions that hav e b een severely criticized by historians, arc haeologists, and sc holars in rel ated disciplines. W e summarize the results of an alter- native computation u sing Bay es’ theorem that estimates a probabilit y of less t h an 2% th at the T alpiot tomb b elongs to Jesus of N azareth. 1. Introdu ction. Andrey F euerv erger notes in his article that assu mp- tions A.1 through A.9 are “not unive rsally accepted.” W e argue that most historians and arc h aeologi sts actually disb eliev e his key assum p tions. (W e agree w ith F euerv erger that the computational metho d he pr op oses can b e extremely us efu l for difficult problems such as the T alpiot tom b .) Assumption A.7 (the largest driver of his results) is almost universally rejected b y schol ars in the relev an t fields. S ev eral other assumptions are extremely du bious, and e ach of them b iases the result tow ard H 1 . Since all statistica l b iases in F euerverger’s RR v alues accum u late multiplic atively , the net effect is an enorm ous bias to w ard H 1 . In this article, we will lo ok first at the m ost egregious p r oblem, the “Mari- amenou” inscription, whic h Simc h a Jacobovic i identified wit h Mary Mag - dalene t hrough a long c hain o f reasoning th at has b een sev erely criti cized b y historians. In less detail, w e w ill examine fiv e other serious p r oblems. By F euerv erger’s o wn accoun t, eliminating t w o of these statistical biases (the t wo relating to Mary Magdalene) is su fficien t to destroy the statistical sig- nificance of H 1 . But all six statistical biases s hould b e eliminated f rom the baseline mo del of the pr ob lem. W e describ e a series of calculations using Ba yes’ theorem that show that the probabilit y that th e tom b b elongs to Jesus of Nazareth is at most ab out 2%, and may b e m uc h less. Received Novem b er 2007; revised January 2008. Key wor ds and phr ases. Jesus of N azareth, T alpiot tomb, Bay es’ theorem, statistical inference, distribu t ion of n ames, onomasticon. This is a n electronic r eprint of the o riginal article published by the Institute of Mathematical Statistics in The Annals of Applie d Statistics , 2008, V ol. 2, No . 1, 8 4–90 . This r eprint differs fro m the original in pa gination and t ypo graphic detail. 1 2 R. IN GERMANSON 2. The primary problem: The “Mariamenou” inscription. On e of the ossuaries b ears an inscription th at is usually translated “Mariamenou [who is also called] Mara.” Simc h a Jacob o vici ( 2007 ) to ok this to b e a v ariant of “Mariamne” and interpreted it as a r eference to Mary Magdalene. That is, he b eliev ed Mary Magdalene w en t by this name and that very few other w omen did. J acob o vici based h is theory on the w ork of Dr. F rancois Bo von. But Bo von ( 2007 ) immediately rep udiated this inte rpretation of h is w ork in a w eb article. Th e k ey p oin t is this statemen t: “I do not b eliev e that Mariamne is the real name of Mary of Magdalene.” Dr. Ric h ard Bauc kham ( 2007 ), a reno wned exp ert in fi rst-cen tur y Jewish names, has analyzed th e “Mariamenou” inscription in detail in a guest b log article. His conclusions are: (1) Grammat ically , “Mariamenou” is the genitiv e case of the r are form “Mariamenon,” a d imin u tiv e endearment deriving fr om the common name “Mariam.” (2) The name is n ot derive d from “Mariamne.” (3) The name is very rare, and no other in stance is foun d in an tiquity . (4) W e ha v e no evidence that Mary Magdalene ever w ent b y this name. One should ask wh at n ame Mary Magdalene wen t b y , according to the data we hav e. Stephen Pfann ( 2007 ) has tabulated th e references to Mary Magdalene in the v arious b o oks of the New T estament , the earliest sources that men tion h er . Sh e is called by the formal n ame “Mariam” four times and by the shorter, m ore intimate form “Maria” 10 times. These are the only names used in th e New T estamen t to r efer to Mary Magdalene. With these facts at h an d , we can answer the f ollo wing qu estion: Assu ming that Mary Magdalene w as actually b uried in ancien t Jerusalem, if one fi nds the inscription “Mariamenou” in th at cit y , what is the probabilit y that it migh t refer to Mary Magdalene? Th e answer is that the inscription is nei- ther more nor less lik ely to refer to Mary Magdalene than to any other Mary of Jerusalem. (There w ere rough ly 8500 other Marys.) This d emolishes Ja- cob o vici’s theory , b ecause “Mariamenou” simply can’t b e id entified as “the real name” of Mary Magdalene. In F euerv erger’s article, he assigns an RR v alue to the Mariamenou in- scription that carries an illicit factor of (1 / 44), du e to his b elief that the inscription “represents the m ost appropriate sp ecific app ellation for Mary Magdalene from among those kno wn.” But it d o esn’t, and therefore this factor (1 / 44) should b e c hanged to 1. This faulty assumption biases the en tire calculat ion ve r y strongly to ward H 1 and is the p rimary driv er b ehind the allegedly remark able results. 3. Five other significan t problems. In addition to the “Mariamenou” issue, there are a num b er of other problems in F eu erv erger’s w ork th at bias DISCUSSI ON 3 the computation tow ard H 1 . Eac h of them con tr ib utes a facto r smaller than 1. The resu lt of multiplying them all together is an enormous bias to wa rd H 1 . These problems are as follo ws: (1) Assumption A.3 asserts that “the most appr op r iate rend ition of th e name of the mother is Mary a.” Note that “Mary a” is the short form of the more formal n ame “Mariam” and is often sp elled “Maria” in English. Assumption A.3 asserts that the mother of Jesus could not b e listed as “Mariam” on her ossuary . With th is assump tion, F euerverger inserts a factor of (13 / 44) in to his RR v alue for the “Maria” in s cription. The problem is that there is no comp elling reason to b eliev e A.3. The New T estamen t data compiled by Stephen Pfann ( 2007 ) sho ws that the mother of Jesus w as called “Mariam” 13 times and “Maria” s ix times. So the data run s coun ter to F euerverger’s assump tion. The mother could b e called by either n ame. F euer verger’s factor of (13 / 44) is illicit and should b e eliminated. (2) Assumption A.3 lik ewise asserts that the short form “Y oseh” is the most appr opriate rendition of the second br other of Jesus, wh ose formal name was “Y ehosef ” lik e his father. T h e New T esta men t refers to this b r other once by the short f orm and once by the long form . A complicating factor here is that an y r andomly chosen “Y ehosef son of Y ehosef ” would b e qu ite lik ely to carry an alternativ e form of the name, so as to distin gu ish b et ween father and son. F euerv er ger inserts a factor of (7 / 46) into h is RR v alue, whic h is to o small, b ecause it is at the minimum of the range of p ossible v alues. The correct v alue s hould lie somewhere b et we en (7 / 46) and 1. (3) An ins cr ip tion “Judah son of Jesus ” ind icates that the J esus bu ried in the tomb had a son. Jewish men of the time we re very likely to b e married and hav e c hildren . But it is pr obable that Jesu s h ad no sons . Recall that Jesus had four b rothers who assumed p ositions of influence in the early Jesus mo vemen t. If a son also existed, h e w ould lik ely ha ve joined his u ncles in a p osition of influen ce and w e would ha v e h eard of him. Sin ce we h a ve n ot, w e can conclude that the probabilit y that Jesus had a son is lower than the pr ob ab ility f or a rand omly selected man of Jerusalem. F euerverge r’s calculatio n fails to accoun t for this. This in s erts a bias into his computation. (4) If the T alpiot tom b conta in ed the family of J esus of Nazareth, would we exp e ct Jesus to b e in it ? Arc haeologist Jo di Magness ( 2007 ) argued from a historical p ersp ective that we s hould not. (But note James T ab or’s rebu ttal ( 2007 ), which argues that the tom b “should not b e dismiss ed .” W e agree that it should n ot b e dismissed, b ut it must stand on its merits.) Magness and T ab or at least agree that the data indicates th at th e b o d y of J esu s w ent missing within da ys after the cr u cifixion. Th e earliest Jesus m o ve men t explained this by asserting that Jesus w as resu r rected, a claim outside the b ound s of scien tific inv estigat ion. If one lo oks for a natur alistic exp lanation, Magness says that muc h the lik eliest one is that Jesus was r eburied in a 4 R. IN GERMANSON simple trenc h grav e lik e other p o or m en of his time. S he argues on sev eral grounds that it is implausible that Jesus wa s buried in a r o c k-cut tomb like the one at T alpiot. F euerverger’s analysis fails to p enalize H 1 on accoun t of this issu e, thereby in tr o ducing another source of s tatistical bias into h is calculatio ns. (5) W ould Mary Magdalene b e buried in the family tom b of J esu s? Ac- cording to Bauc kham ( 2007 ), the usual pr actice w as that only family mem- b ers we r e b uried in a family tomb. It is p ossible th at Mary Magdalene was a family memb er. It is ev en p ossible that she was married to Jesus . Bu t we can ha ve no certain t y that she w as. Most historians w ould estimate a prob- abilit y substantial ly less than 1 for these p ossibilities. F euerv er ger’s analysis assumes that Mary Magdalene should b e in the tom b and his computation ac hiev es s tatistica l significance only i f she is assume d to b e in the tomb . T his in tro duces another very serious source of statistical bias in to his computa- tions. 4. A calculation usin g Bay es’ theorem. It is b ey ond the s cop e of this short comment to giv e f ull details on a more correct calculation. This jour nal has giv en us space on its w eb site for a 29 page article that defines the statistica l issu es of the tom b and th en describ es a series of calculations w e ha ve p erform ed. Here, we will merely su mmarize the resu lts of that article [Ingermanson ( 2008 )]. W e defin e the t wo ev en ts J and T as follo ws: J = the “Jesus son of Joseph ” in the T alpiot tomb refers to Jesus of Nazareth , T = the observ ation of the rest of the T alpiot tom b data. W e denote the n egation of the even t J by the sym b ol ∼ J . W e are in terested in computing the conditional probabilit y P ( J | T ) u sing Ba y es’ theorem: P ( J | T ) = P ( T | J ) P ( J ) P ( T | J ) P ( J ) + P ( T | ∼ J ) P ( ∼ J ) . Define the t w o ratios α ≡ P ( ∼ J ) P ( J ) , β ≡ P ( T | ∼ J ) P ( T | J ) . Then our formula simplifies to P ( J | T ) = 1 1 + αβ . DISCUSSI ON 5 The r esults of man y computations can b e summ arized as follo ws: α tend s to b e large, while β is n ear 1. Therefore, P ( J | T ) tends to b e small. W e can estimate α very quickly . F euerve rger q u otes the resu lts of Camil F u c hs ( 2004 ) that th e num b er of adult males w h o died in Jerusalem in the r elev an t time p erio d was ab out 36420. Th is is o verly pr ecise, but it is reasonable in m agnitude. Assuming that 4% of men we re n amed Jesus and 8.8% were named Joseph , w e estimate the n u m b er of men n amed “Jesus son of J oseph” to b e ab out 128. On e of these m en was Jesus of Nazareth. Th e other 127 are un kno wn to history . Therefore, if w e are giv en a rand omly c h osen man of Jerusalem n amed “Jesus son of Joseph,” the probab ility that he is Jesu s of Nazareth is P ( J ) = 1 / 128. T he probability th at he is n ot is P ( ∼ J ) = 127 / 128. T aking the ratio, w e estimate α ≈ 127. In general, if there were N J men of Jeru salem named “Jesus son of Joseph ,” then we hav e α = N J − 1. The estimation of β is muc h more complicated and we describ e it in detail in the supplemental article [Ingermanson ( 2008 )]. The general pro cedur e is as follo ws: W e are comparing t wo hyp otheses, J and ∼ J , usin g the data T to dis- tinguish b et ween the t wo. F or eac h of th ese tw o hyp otheses, w e imagine a statistica l en s em ble of tombs “similar” to the T alpiot tom b . W e’ll make ran- dom dra ws f rom eac h ensemble an d tabulate the frequ ency of “hits” (random dra w s that agree w ith the data T ) . W e’ll stipulate th at ea c h mem b er of these t wo ensembles s hould con- tain an ossu ary inscrib ed w ith “Jesus son of Joseph” and a second ossuary inscrib ed with “Judah son of Jesus.” It should also con tain tw o ossuaries b earing female n ames, tw o ossuaries b earing m ale names, and four u nin- scrib ed ossuaries. The distribu tion of names on the in scrib ed ossu aries m u st matc h the distribu tion of the names of p ersons living in J erusalem in the first cen tury , su b ject to the constrain ts of the tw o hyp otheses. In the case of the ∼ J hyp othesis, there are no constraints. In the case of the J hyp othesis, the only constraint is that the tom b m u st contai n at least the names of certain memb ers of the family of Jesus, with any remaining slots in the tomb filled with names c hosen using the distribution of names in Jerusalem. The pro cedure outlined a b o v e is similar in spirit to that follo w ed by F euer verger. Here are the primary differences in our calculat ions. W e sa y that: (1) The name of the mother of Jesus could ha ve b een ins crib ed as any form of Mary , including “Mary a,” “Mariam,” or an y other v arian t (ev en including the muc h-debated “Mariamenou Mara” inscrip tion). 6 R. IN GERMANSON (2) “ Judah son of Jesus” is considered less lik ely to app ear in the tom b of Jesus of Nazareth than in the tomb of a randomly selected “Jesus son of Joseph.” (3) Jesus of Nazareth is considered less likely to b e bu ried in a ro ck-cut tom b than wa s a r an d omly selected “Jesus son of J oseph .” (4) Ma r y Magdalene is n ot assumed to b e in the tom b , and the “Maria- menou Mara” in scription is not assu m ed to b e an app ellatio n that applies to her with any higher probabilit y than to any other Mary of Jeru salem. (5) The probabilit y of find ing a Y oseh in the tom b is rec koned to b e higher than usual, b ecause the patriarch of the T alpiot family is named Joseph. (6) The measure of “surpr isin gness” is the count of family memb ers in the tom b, not F euerv erger’s RR v alues. W e use six d ifferen t wa ys of d efining this count. The calculation w as p erformed in Ja v a us ing a wide v ariet y of assump tions for the comp osition of a “Jesus family tom b” and using six d ifferen t defi- nitions of “surpr isingness.” R an d om dr aws w ere made in groups of 10,000, and results were tabulated. The baseline calculati on retur n ed an estimate for the upp er b ound of P ( J | T ) at ab out 2% (with a standard deviation of ab out 2%). A num b er of v arian ts were tried, and the h ighest v alue foun d for P ( J | T ) w as 5.67%, u sing one assump tion w e consider un lik ely . (The assum ption that Y oseh sh ould b e exactly as rare in the T alpiot tom b as it is in tom b s that do not ha ve a patriarc h named Joseph.) W e found that by tighte ning t wo assumptions, the upp er b ound could b e substanti ally redu ced. These are as follo ws. W e hav e assumed that the r elativ e p robabilit y ρ son that Jesu s had a son (as compared to other men of his time) was less than 1. Th at is, we d efined a r andom v ariable ρ son uniformly distributed on the in terv al [0 , 1]. Many historians wo u ld argue th at th is distribution should b e s tr ongly we igh ted to wa rd zero. Doing so would strongly reduce our estimates of P ( J | T ) . Lik ewise, we hav e assum ed that the relativ e p robabilit y ρ tom b that Jesus w as r eb uried in a ro ck-c u t tom b (as compared to other men of his time) w as also less than 1. W e defined a random v ariable ρ tom b uniformly d istributed on the interv al [0 , 1]. As noted earlier, Jo di Magness ( 2007 ) has argued strongly that ρ tom b should b e hea vily w eigh ted to wa rd zero. Doing so would again sharply redu ce our estimates for P ( J | T ). W e lea v e it to h istorians and arc h aeologi sts to debate suc h matters. W e exp ect that their conclusions will tend to reduce our upp er b ound f or P ( J | T ) to b e less than 2%, b u t it is imp ossible to predict how far it migh t drop. Suc h matters are irr educibly sub j ectiv e. DISCUSSI ON 7 5. Conclus ion. F euerverger’s computation cont ains a num b er of statis- tical biases, eac h of wh ic h fa vors H 1 . One of these (the “Mariamenou” in- scription) introd uces an illicit factor of 1 / 44 to RR, wh ic h accoun ts for a v ery strong bias all b y itself. But fi v e other factors en ter in with m o derate statistica l b ias to ward H 1 , and the net effect is to create the app earance of statistica l significance where none actually exists. W e h a ve p erf orm ed a s eries of calculations usin g Ba yes’ theorem that estimate a lik ely u pp er b ound for th e pr obabilit y that the T alpiot tom b is the tom b of Jesus of Nazareth. This upp er b ound is ab out 2% with a standard deviation of ab out 2%. Ac kn owledgmen ts. I thank Ja y Cost for readin g this article and making commen ts. I also thank Richard Bauc kham, Mark Go o d acre, Gary Hab er- mas, Mic hael Heiser, S tephen Pfann, and James T ab or for helpful discussions o ve r the last several mon ths . I tak e full credit f or an y errors. SUPPLEMENT AR Y MA TERIAL Analysis of the T alpiot tom b usin g B a yes’ Theorem and r andom v ari- ables (doi: 10.121 4/08-A OAS99G SUPP ; .p df ). W e analyze the T alpiot tom b, whic h has b een alleged to b e the family tom b of Jesu s of Nazareth. Using Ba y es’ Th eorem, we d eriv e a simple function that estimates the probabilit y that the tom b h ouses the remains of Jesu s and his family . Unfortun ately , this fu nction cannot b e ev aluated exactly , b ecause several of the key param- eters are unk n o wn . By us ing random v ariables with r easonable p robabilit y distributions, we examine the mean b eha vior and r ange of the fu nction un- der a v ariet y of conditions. W e conclude that th e probability is lo w (on the order of 2% or less) that the T alpiot tomb is the family tom b of Jesu s of Nazareth. REFERENCES Bauckham, R. (2007). Guest article on Chris Tilli ng’s b log on Marc h 1, 2007. Av ailable at http://www .christil ling.de/blog/2007/03/guest- post- by- richard- bau ckham.html . Bov on, F. (2007). Article on the Society of Biblical Literature web site, p osted Marc h, 2007. A v ailable at http://sbl - site.org/Article .aspx?ArticleId=656 . Fuchs, C. ( 2004). Demo cracy , literacy and n ames distribution in ancient Jerusalem— how many James/Jacob son of Joseph, brother of Jesus were there? Polish J. Bibl ic al R ese ar ch 1–30. Ingermanson, R . (2008). Supp lement to “Discussion of: Statistical analysis of an arche- ological fi nd.” DOI: 10.1214/08 -AO AS 99GSUPP . Ja cobo vici, S . and Pellegrino, C. ( 2007). The Jesus F am ily T omb . Harp erOne, S an- F rancisco. Magness, J. ( 2007). Art icle on th e So ciet y of Biblical Literature w eb site, p osted March, 2007. A v ailable at http://sbl - site.org/Article .aspx?ArticleId=640 . Pf ann, S. (2007). Av ailable at http://www .uhl.ac/L ost_Tomb/ HowDoYouSolveMaria/ . 8 R. IN GERMANSON T abor, J. (2007). Article on the So ciety of Biblical Literature web site, p osted Marc h, 2007. A v ailable at http://sbl - site.org/Article .aspx?ArticleId=651 . 2210 W. Main S treet Suite 107, Box 103 Ba ttle Ground, W ashington 9860 4 USA E-mail: randy@rsingermanson.com URL: www.Ingermanson.com