Discussion of: Bayesian views of an archaeological find

The Annals of Applie d Statistics 2008, V ol. 2, No. 1, 97–98 DOI: 10.1214 /08-A OAS99B Main articl e DO I: 10.1214/ 08-AOAS99 c  Institute of Mathematical Statistics , 2 008 DISCUSS ION OF: BA YESIAN VIEWS OF AN AR C HAEOLOGICAL FIND By Joseph B. Kadane Carne gie Mel lon University Andrey F euerv erger ( 2008 ) is to b e congratulated on ha ving giv en u s suc h a careful analysis of a v ery interesting data set. He h as obviously gone to great efforts to understand the archaeo logy (in sev eral languages) and bac k- ground of the tom b in question, and the literature and history surrou n ding it. This effort is exactly wh at a mo dern statistici an should b e doing in an applied problem. Unfortunately F euerverger is hamp ered, in m y view, b y his pr edisp osition to wa rd sampling theory . His tec h nique relies on his R R -v alues (“relev ance and rareness”), but h e giv es n o theory of RR. Ju st what is it? What justifies m u ltiplying them toget her? What hav e y ou got wh en yo u ’re done? Second, his metho d computes the pr obabilit y of data as or more extreme than that observ ed w ere th e null hyp othesis true, which violates the lik eliho o d pr in ci- ple b ecause it coun ts as relev an t d ata th at did n ot o ccur . Finally , his metho d is v ery limited in the conclusions it p ermits one to d ra w: either the null h y - p othesis is false or something u n usual has happ ened. W ell, wh ic h is it? Usin g his paradigm, he is u nable eve n to giv e a probability of whic h of these is the case. A great deal of effort go es into establishing a conclusion whose form do es not add ress the question of in terest, at least as I in terp ret it. By cont rast, a Ba y esian treatmen t has clear-cut an d simple ru les. These ha ve b een w orke d out extensively for pr oblems in forensic science; indeed the present problem can b e so r egarded. Th e qu estion, as F eu er verger himself p oints out, is to calculate P ( B | A ) /P ( B | A ) where A is the ev ent that the T alpiyo t tom b is that of the N T family , and A is that it is not. The even t B is the evidence w e ha v e, n amely the sp ecific n ames found in T a lpiy ot. P ( B | A ) is probabilit y of this tom b arising if it we re the tom b of the N T family . Th us it inv olv es what other renditions of names migh t ha v e b een us ed for the p ers ons in the N T family , and the p ossib le identiti es of the unidentified p ersons in the tomb. S imilarly P ( B | A ), whic h is essent ially what he is Received Octob er 2007; revised December 2007. This is an electro nic reprint of the or iginal a rticle published b y the Institute of Mathematical Sta tistics in The Annals of Applie d Statistics , 2008, V ol. 2, No. 1, 97–9 8 . This reprint differs from the original in pag ination and t ypo graphic detail. 1 2 J. B. KA DANE computing from the onomasticon, is the pr ob ab ility of this configuration arising fr om some other family or grou p of p eople. While he says that this sp ecification of B is “a wkward to work with,” it seems to me that it leads us to addr ess th e essentia l questions in analyzing the T alpiyo t tom b. H¨ ofling and W asserman ( 2008 ) and Ingermanson ( 2008 ) in preceding com- men ts on the pap er giv e differing Bay esian analyses of this p roblem, and Mortera and Vicard ( 2008 ) stated how they w ould u se DNA analysis in one. Should we b e distur b ed that the former tw o make differen t assumptions, and deriv e differen t p osterior p robabilities? I w ould argue not. The strength of the Ba y esian appr oac h is th at it requir es the assump tions to b e stated ex- plicitly an d argued for. The acceptabilit y of th ose assumptions is for eac h reader to judge for himself or herself. All the Ba yesian argument ens u res is that eac h writer is coheren t, that is, d o es not con tain in ternal contradic- tions in a certain tec hnical sense. Th us the Ba yesia n view of probabilit y is arguably like a language. That a sentence is in grammatical En glish do es not require the reader to agree with it; prop er grammar only helps us to understand what the writer means. Similarly , an opinion expressed in p rob- abilistic terms is explicit, th at is, a reader can understand what the writer’s view is, bu t it is up to the wr iter to b e p ersuasiv e to the reader. Eac h reader, then, needs to state the b eliefs foun d m ost congenial, and to compu te h is or her o wn p osterior probab ility accordingly . Finally , it is obviously necessary to sa y something ab out h o w the statis- tical analysis of this data set relates to the religious b eliefs of man y p eople. F ortunately there is no con tradiction b et w een the Bay esian p aradigm and suc h b eliefs. Bay es Theorem in o d ds form reads, as F e uerv erger p oints out, P ( A | B ) P ( A | B ) = P ( A ) P ( A ) × P ( B | A ) P ( B | A ) . Here the factor P ( A ) /P ( A ) is the prior o dd s of the even t A . F or those whose religious b eliefs sp ecify P ( A ) = 0 and P ( A ) = 1 (i.e., there is no c hance that the tom b is that of the N T family), w hatev er the lik eliho o d contribution [here P ( B | A ) /P ( B | A )], the p osterior od ds of A [here P ( A | B ) /P ( A | B )] are zero. T his set of b eliefs is coheren t in th e tec hnical sen s e (i.e., it do es not lead to sure loss), and hence is f ully consistent with the Ba y esian view. REFERENCES Feuer ver ger, A . (2008). St atistical analysis of an arc heological find. Ann. Appl. Statist . 2 3–54. H ¨ ofling, H. and W asserman, L. (2008). Discussion of: “Statistical analysis of an arc he- ological fin d” by A. F euerverger. Ann. Appl. Statist. 2 22–83. Ingermanson, R. (2008). Discussion of: “Statistical analysis of an arc heological find” by A. F euerver ger. Ann. Appl. Statist. 2 84–90. DISCUSSI ON 3 Mor tera, J. and V icard, P. (2008). Discussion of: “Statistical analysis of an arc h eolog- ical find ” by A. F euerv erger. Ann. Appl. Statist. 2 91–96. Dep ar tment of St a tistics Baker Hall 232A Carnegie Mellon University Pittsburgh, Pennsyl v ania 15213 USA E-mail: k adane@stat.cm u.edu