Mathematical Proof Jesus Is The Son of God



1
"He Is Not Here," Or Is He?
A Statistical Analysis of the Claims Made in The Lost Tomb of Jesus
By Jay Cost and Randy Ingermanson1
Introduction
In March, 2007 Emmy-winning filmmaker Simcha Jacobovici presented a
documentary, The
Lost Tomb of Jesus, that proffered a startling thesis: Jesus of
Nazareth, whom most Christians
believe rose bodily from the dead, was actually buried in a tomb in
the present-day East Talpiot
neighborhood of Jerusalem.2
As evidence for the thesis, Jacobovici enlisted the aid of University
of Toronto professor of statistics
Andrey Feuerverger, who, Jacobovici claimed, calculated the odds that
this is the Jesus
family tomb at 600 to 1.3
In the present paper we offer a critique of this statistic and an
alternative estimate. We argue that
Bayes' Theorem is the optimal way to calculate the likelihood that
this tomb belonged to Jesus of
Nazareth. We further argue that the estimate derived from Bayes'
Theorem is fairly insensitive
to a series of assumptions that can be altered as the vital debate
among historians and archaeologists
requires. Finally, making what we believe to be a set of reasonable
assumptions, we offer a
series of estimates of the likelihood that the Talpiot tomb belonged
to Jesus of Nazareth.
1 We are indebted to a number of people for useful discussions on this
topic. We gratefully acknowledge the input of
Steven Avery, Joe D'Mello, Mark Goodacre, Gary Habermas, Michael
Heiser, John Koopmans, Stephen Pfann,
John Poirier, Chris Rollston, James Tabor, David Tyler, and Ben
Witherington, III.
2 See Simcha Jacobovici. The Lost Tomb of Jesus. Discovery Network.
First Aired March 4, 2007.
3 In the aftermath of the documentary's release, Jacobovici and the
Discovery Channel modified their interpretation
of Feuerverger's mathematical computations. It is unclear to us
exactly what Jacobovici now believes that
Feuerverger's number demonstrates. However, it is nevertheless clear
that he believes now, as he did before the
updates made to the Discovery Channel's documentation, that
Feuerverger's number provides some analytical "purchase"
on whose tomb was discovered at the Talpiot site. In other words,
while we are not sure the exact nature of
the inference that Jacobovici draws about the tomb from the number, we
are sure that he is drawing some kind of
inference about the tomb ownership from the number. We disagree even
on this minimal point. We think that this
number is insufficient for any inference about the tomb, and we will
justify this disagreement in the course of this
essay.
2
The Talpiot Discovery and the Documentary's Argument
In 1980, the Israeli Antiquities Authority excavated a tomb in East
Talpiot, a Jerusalem neighborhood.
Within the tomb, archaeologists discovered ten ossuaries, or bone
boxes used to intern
the deceased between the first century BCE and first century CE.4 Six
of the ossuaries had
names inscribed upon them - five in Aramaic and one in Greek. The
Aramaic inscriptions, when
translated into English, were "Judah son of Jesus," "Jesus son of
Joseph," "Jose" (a diminutive of
"Joseph"), "Mary" and "Matthew." Kloner (1996), following Rahmani
(1994), translates the
Greek inscription as "Mariamne [also called] Mara."5
In the documentary The Lost Tomb of Jesus, Jacobovici argues that four
of these names correspond
with the family of Jesus of Nazareth. "Jesus son of Joseph" might have
been a way to address
Jesus of Nazareth. Further, Jesus of Nazareth had a brother named Jose
and a mother
named Mary. Jacobovici further speculates that Kloner (1996) and
Rahmani (1994) have mistranslated
the Greek inscription. He asserts that the correct translation is
"Mary [also called] the
Master." He goes on to argue, using the 4th century apocryphal
document called The Acts of
Philip, that this is a name by which Mary Magdalene might have been
known.
It is from this cluster of names that Professor Feuerverger derives
his figure. To date, Professor
Feuerverger has not submitted a formal paper for either public viewing
or scholarly vetting.6
Accordingly, his statistical intuitions must be gleaned from the
popular consumption book, The
Jesus Family Tomb,7 and the documentary's website.8 The formula
offered in the latter is as
follows:
4 See Craig Evans. Jesus and the Ossuaries. Waco, TX: Baylor
University Press, 2003.
5 See Amos Kloner. "A Tomb with Inscribed Ossuaries in East Talpiyot,
Jerusalem," Atiquot 29, 1996, and L.Y.
Rahmani. A Catalogue of Jewish Ossuaries in the Collections of the
State of Israel. Israel: Israeli Antiquities
Authority / Israel Academy of Sciences and Humanities, 1994.
6 Currently, Feuerverger has made an explanation of some of the
assumptions inherent to his calculation available on
line. See Andrey Feuerverger. "The Tomb Calculation, Letter to
Statistical Colleagues. March 12, 2007."
http://fisher.utstat.toronto.edu/andrey/OfficeHrs.txt. Accessed March
26, 2007. Note that Jacobovici and Pellegrino
(2007) indicate that Feuerverger has submitted a paper to scholarly
review, but the above-referenced letter from
Feuerverger indicates that this is not the case. Personal
correspondence between Feuerverger and one of the authors
confirmed that Feuerverger's letter is correct on this point.
7 See Simcha Jacobovici and Charles Pellegrino. The Jesus Family Tomb.
San Francisco: HarperCollins, 2007.
8 See "The Lost Tomb of Jesus: The Discovery Channel."
http://dsc.discovery.com/convergence/tomb/tomb.html.
Accessed March 26, 2007.
3
Initial Computation
Jesus Son of Joseph:
1 in 190
Mariamne:
1 in 160
Matia:
1 in 40
Yose:
1 in 20
Maria:
1 in 4
= 1 in 97,280,000
Second Computation
(Matthew Eliminated)
Jesus Son of Joseph:
1 in 190
Mariamne:
1 in 160
Yose:
1 in 20
Maria:
1 in 4
= 1 in 2,400,000
Third Computation
(Unintentional Historical Biases Factored
In)
2,400,00 / 4 = 1 in 600,000
Fourth Computation
(All Potential Tombs Factored In)
600,000 / 1,000
= 1 in 600 = "Probability
Factor"
Critique of the Statistical Method
The above method, as we are able to interpret it, suffers from several
important defects that prevent
any valid inference about the family in the tomb from being drawn.
These defects can be
grouped into two types: theoretical and computational.
Theoretical Defects
This method generally lacks probative value. That is, the "probability
factor" does not speak to
the likelihood that Jesus of Nazareth was buried at this tomb. It is
simply an estimate of the
probability of finding these four names - "Jesus son of Joseph,"
"Mariamne," "Jose" and
"Maria" - in all tombs with exactly four inscribed names. It is does
not speak to whether this
family is actually the family of Jesus of Nazareth. This is because it
fails to consider two important
factors.
It does not consider the unlikelihood of ever finding Jesus of
Nazareth's tomb in the first place.
It might be a unique occurrence to find these four names, but the
uniqueness of this occurrence
4
must be understood in the context of the uniqueness of finding Jesus
of Nazareth at all. For instance,
if there were a 1 in 1 billion chance that we would find Jesus of
Nazareth's ossuary, the
probative value of the unadjusted probability in the above table,
which shows a 1 in 2.4 million
likelihood, would be severely diminished. Thus, this other probability
must be taken into account
before any inferences can be drawn. To fail to do so is to commit what
is known as the
prosecutor's fallacy. The unlikelihood of finding the particular
evidence that has been found
must be weighed against the unlikelihood of making the match that the
documentarians wish to
make.
Second, it assumes that there is no doubt that Jesus of Nazareth - if
he was buried in a tomb -
would certainly be buried with these individuals. The implication here
is that this is a "unique"
match with the family of Jesus of Nazareth. However, this hypothesis
is underdetermined. Even
if we accept the uniqueness of the names, and there is good reason not
to, there are reasons to
expect Jesus of Nazareth to be buried with another cluster of names.
This estimation, further,
cannot be taken in the context of the four matches. It must, rather,
be taken in the context of all
six matches - for this "Jesus son of Joseph" was not interned with
three other known people. He
was interned with five. Thus, we should inquire whether there is
reason to expect that Jesus of
Nazareth would be interned with these five names.
Computational Defects
The calculation itself also suffers from four computational problems.
First, there are six inscribed ossuaries in the tomb. If the intention
is to discover the probability
of finding these four names in a given tomb of this type, it should be
calculated with a mind to
the number of inscribed ossuaries in the tomb and not to the number of
"matches" that one
wishes to make. For instance, if there are six inscribed ossuaries,
the chances of finding these
four names should be calculated not as the chance of finding the four
names in four attempts, as
is the case here, but of finding these four names in six attempts.
5
Second, the factor that "adjust[s] for unintentional biases in the
historical record" seems to be ad
hoc. This, it is explained in Jacobovici and Pellegrino (2007), is
intended to account for the
family members who might have been placed at the tomb but are in fact
not there - namely the
brothers James, Simon and Jude. However, it is unclear to us why a
factor of four is sufficient
for dealing with the "absences" from the tomb. It seems more
appropriate to include the factor
we mentioned above - an evaluation of the likelihood that Jesus of
Nazareth would be interned
with this particular cluster of names. Hopefully, at a future date
Professor Feuerverger will have
the opportunity to clarify his justification.
Third, the documentarians hypothesize that Mary Magdalene was interned
in one of the ossuaries,
and that "Mariamne [also called] Mara" is a mistranslation that, when
corrected, points toward
her. However, they do not offer a convincing argument as to why this
is expected to be the
case. At best, they provide a potential connection between the two.9
This connection should be
critically evaluated before the name "Mariamne [also known as] Mara"
is included in the computation.
Absent this inquiry, the statistical computation should be limited to
the three known
matches between the tomb and Jesus of Nazareth's family. Including
"Mary Magdalene" in the
statistical analysis heavily biases the probability estimate toward
the hypothesis.
Fourth, the computation fails to account for the potential bias that
occurs with the consideration
of the name "Jose," which Jacobovici takes as a rare and highly
significant reference to Jesus of
Nazareth's brother. It fails to consider that "Jose" has been found
given the discovery of a reference
to "Joseph." It is reasonable to believe that "Jesus son of Joseph"
and "Jose" might not be
independent events - namely, that "Joseph" might have generally been a
family name. Accordingly,
when a Joseph is found in a family, one is more likely to find another
Joseph, or a person
with a variant of the name, such as "Jose." Thus, we should expect P
(Jose|Joseph) > P(Jose).
Unfortunately, the historical record is not sufficiently detailed to
offer conditional probabilities
such as the one that we have in mind. However, this is not to say that
the computation's incorporation
of "Jose" is optimal. While it might be impossible to estimate the
precise difference be-
9 For this argument, they rely upon the research of François Bovon.
However, in a recent article, Bovon argues: "I
do not believe that Mariamne is the real name of Mary of Magdalene.
Mariamne is, besides Maria or Mariam, a possible
Greek equivalent, attested by Josephus, Origen, and the Acts of
Philip, for the Semitic Myriam." See François
Bovon. "The Tomb of Jesus." The Society of Biblical Literature Forum.
http://sblsite.
org/Article.aspx?ArticleId=656. Accessed March 26, 2007.
6
tween P(Jose|Joseph) and P(Jose), it seems intuitive to expect that
the former is greater than the
latter. Thus, Jacobovici and Feuerverger are likely wrong to take the
name "Jose" to occur with
less frequency than the name "Joseph" (which they do). They should,
instead, expect a "Jose" or
some variant to occur in a tomb like this, i.e. one that was used by a
family known to have a "Joseph"
within it, with a greater likelihood than a "Joseph" would in the
general population.
Even with these critiques, the above figure represents an important
step in calculating the probability
that this tomb belongs to Jesus of Nazareth. Jacobovici and
Feuerverger are to be credited
for astutely recognizing that the other names in the tomb are the key
components in determining
whether the Talpiot tomb once held the remains of Jesus of Nazareth.
Building upon their intuition,
we propose a different, more comprehensive approach than they have
adopted.
Bayes' Theorem and the Jesus Tomb
Introduction
We propose the use of Bayes' Theorem to evaluate the likelihood that
this tomb belongs to Jesus
of Nazareth. Indeed, it is our intuition that Bayes' Theorem offers
the most direct way to answer
the question at hand. We also believe that Jacobovici's argument is
fundamentally Bayesian.
Jacobovici is interested in the extent to which the other names in the
tomb provide a clue to the
person interned in the "Jesus son of Joseph" ossuary. In other words,
his endeavor is to find a
conditional probability: given the other names in the tomb, what is
the likelihood that Jesus of
Nazareth is interned in the "Jesus son of Joseph" ossuary, what we
define as P(J|T). Meanwhile,
it is Jacobovici's intuition that the Talpiot tomb provides a close
match with what we would expect
in Jesus of Nazareth's tomb. That is, he believes that what we define
as P(T|J) is very large
- i.e., if we were assuredly to find Jesus of Nazareth's tomb, we
would expect many of these
names within it.
What Jacobovici has done, then, is define one conditional probability,
P(J|T), in terms of another,
P(T|J). This is a Bayesian argument. Unfortunately, as the last
section makes clear,
7
Feuerverger's statistical analysis does not speak to this. It suffers
from two theoretical problems:
(i) any inferences drawn from his number will run afoul of the
prosecutor's fallacy;
(ii) it implicitly assumes an artificially high value of P(T|J).
What is required, then, is a reformulation and formalization of what
we take to be Jacobovici's
accurate intuitions.
Statement of the Problem
A tomb has been found in East Talpiot, Jerusalem, containing ten
ossuaries. Six of these were
inscribed with names, while four were not. The excavating
archaeologists estimated that the
tomb might have held up to 35 bodies. This was not based on any count
of bones within the
tomb. Rather, it was an estimate based on many similar tombs around
Jerusalem dating from the
same era, the first century.10
Two of the ossuaries are special and indicate a rudimentary family
tree. One is inscribed "Judah
son of Jesus" and the other is inscribed "Jesus son of Joseph." While
there is no proof that the
"Jesus" in these two inscriptions is the same man, that is the general
assumption, since a tomb
like this generally contained extended families over several
generations.
The immediate question to ask is whether the ossuary inscribed "Jesus
son of Joseph" might
have contained the bones of Jesus of Nazareth. The ossuary dates to
the first century and the
tomb lies within a few miles of the crucifixion site of Jesus of
Nazareth. What is the probability
that Jesus of Nazareth might be the Jesus of the tomb?
If that were all the information we had, then the answer would be
fairly easy to compute. Suppose
there were
EURO
NJ men in Jerusalem named "Jesus son of Joseph." The probability of
choosing
any one of these at random is clearly 1/
EURO
NJ .
10 See Kloner (1996).
8
We note that Jesus of Nazareth is not a randomly selected "Jesus son
of Joseph" from ancient
Jerusalem. We have more information about him than we do about the
other men named "Jesus
son of Joseph."
Most scholars accept the New Testament account that Jesus was buried
immediately after his
crucifixion in a tomb owned by Joseph of Arimathea, roughly three
miles from the East Talpiot
location of the alleged "Jesus family tomb." Nobody believes Jesus of
Nazareth is still in this
original resting place.
Christians generally believe that Jesus was physically resurrected
about a day and a half after his
death and that he eventually ascended to heaven. If that is not the
case, then he was instead reburied
elsewhere (perhaps a tomb such as this, or a common burial pit, or
something else). Conceivably,
he could have been transported some three miles away to the East
Talpiot tomb, but
that is not a sure thing. But neither is it a sure thing that he was
not.
The question of whether Jesus ascended to heaven is a faith-question,
which will be answered
differently by different people. The question of whether Jesus might
have been transported three
miles to East Talpiot for reburial is a historical question, which
will be answered differently by
different historians. In either case, there is a subjective element.
This appears to put us at an impasse. But does it really? We argue
that it does not. The subjective
element introduces uncertainty into the calculations, but science has
well-defined ways for
dealing with numerical uncertainties. Thus, while we might not be
capable of identifying an expected
probability value, we might nevertheless be able to identify the range
of values that this
probability might take.
There are several points in this calculation at which we will run into
"judgment calls" that depend
on subjective elements. We will encapsulate each of these points in a
"fuzzy factor" that is
strictly bounded. We will then carry out the calculations and show
that the final range of probabilities
is still bounded. We will now define the first "fuzzy factor."
9
Define
EURO
F1 = "the probability that the body of Jesus of Nazareth was
transported to a tomb of the
type found at East Talpiot and featured in the documentary The Lost
Tomb of Jesus."
Obviously,
EURO
F1 is a subjective factor. Those who believe with certainty in a
physical resurrection
and ascension of Jesus will assign
EURO
F1 a value of zero. Those who believe with certainty that Jesus
was reburied in a trench grave near the site of the crucifixion will
likewise assign
EURO
F1 a value
of 0. Those who are willing to consider the possibility that he was
transferred a few miles away,
and that he might have ended up in a family tomb, will assign
EURO
F1 a non-zero value. Those who
consider the tomb a bad demographic or socio-political "match" for
Jesus of Nazareth's family -
for instance because the tomb might be a "middle class" tomb, etc -
would assign a low non-zero
value. Those who see the Talpiot tomb as a good potential match would
assign it a higher number.
EURO
F1 cannot be less than 0 or greater than 1, since it is a probability.
So
EURO
F1 is a bounded parameter,
and it is likely to have only limited effect on our calculations. We
can run calculations
with different values of
EURO
F1 when we complete our analysis to see what is the range of effects
it
can have.
We asked earlier for a first estimate of the probability that Jesus of
Nazareth is buried in the East
Talpiot tomb. We need some notation. Define:
J = The event that Jesus of Nazareth was buried in the East Talpiot
tomb.
~J = The event that Jesus of Nazareth was not buried in the East
Talpiot tomb.
P(J) = The probability that Jesus of Nazareth was buried in this tomb.
P(~J) = The probability that Jesus of Nazareth was not buried in this
tomb.
Obviously, since J and ~J are exclusive events, P(J) and P(~J) add up
to 1.
Then we have the following naïve estimates of P(J) and P(~J):
EURO
P(J) =
F1
NJ
P(~ J) =1- P(J)
We note that
EURO
NJ has been estimated by different scholars to be anywhere from a few
hundred up
to about 1,000.
10
If this were all the information we had, then the tomb would not be
interesting since P(J) is very
small. However, the tomb also contains ossuaries bearing the names
"Mary," "Mary," "Matthew,"
and "Jose." We know that Mary was the name of the mother of Jesus of
Nazareth and the
name of several other women in the close circle of Jesus (including
Mary Magdalene, Mary of
Bethany, and probably others). Furthermore, one disciple of Jesus was
named Matthew and one
of his brothers was named Jose. These factors tend to strengthen the
case that the tomb might
belong to Jesus of Nazareth.
However, there is also some counter-evidence. One ossuary in the tomb
bears the name "Judah
son of Jesus." No record tells us that Jesus of Nazareth had any
offspring at all, much less a son
named Judah. We should note that no historical records explicitly deny
that Jesus had children,
but we would expect to have heard of progeny if they existed. Three
leaders in the earliest Jesus
community were his brothers James and Judah and his cousin Simon. This
indicates that the
family of Jesus played an important role in the early community. A son
would have played a key
role in this community had he existed. If Jesus had a son, history is
stunningly silent about it, for
no apparent reason.
So the problem then is the following: In light of these additional
five inscriptions, how does the
naïve estimate P(J) change? Will the "pro" evidence outweigh the "con"
evidence?
We will use a few simple techniques from statistics and probability
theory to estimate a revised
probability. A key element will be Bayes' Theorem. We will briefly
review Bayes' Theorem
next.
Bayes' Theorem
In probability theory, one often needs to compute a conditional
probability of one event, given
that another has occurred. This allows one to make inferences based on
partial information.
As an example, suppose two dice are thrown. The odds of rolling a 12
are quite low: 1/36.
However, if we are given information on the state of one of the dice,
we can make a revised es11
timate. If we are told that one of the two dice came up a six, then
the odds of having rolled a 12
are now revised to 1/6. However, if we are told that one of the dice
came up a three, then the
odds of having rolled a 12 are now revised to 0.
Let
EURO
E1and
EURO
E2be two events. We are interested in the probability that
EURO
E1 has occurred, given
that
EURO
E2 has occurred. This is called the conditional probability
EURO
P(E1 | E2) and is defined to be:
EURO
P(E1 | E2) =
P(E1 & E2 )
P(E2 )
One formulation of Bayes' Theorem then follows immediately:
EURO
P(E1 | E2)P(E2) = P(E1 & E2 ) = P(E2 | E1)P(E1)
We can rewrite this as:
EURO
P(E1 | E2) =
P(E2 | E1)P(E1)
P(E2 )
This is a powerful form of the theorem, but we will expand the
denominator by noting that either
EURO
E1 will occur or it will not. So it is easy to prove that
EURO
P(E2) = P(E2 | E1)P(E1) + P(E2 |~ E1)P(~ E1)
Then we get the fairly complicated-looking equation:
EURO
P(E1 | E2) =
P(E2 | E1)P(E1)
P(E2 | E1)P(E1) + P(E2 |~ E1)P(~ E1)
This is useful when we already have a naïve estimate
EURO
P(E1) and we want to improve it in view
of the observation of a second event
EURO
E2. Bayes' Theorem tells us how to make this improvement.
In the previous section, we made a naïve estimate
EURO
P(J) and we then asked how to improve this
estimate based on a set of other observations from the tomb. We will
denote the collection of
these other observations by the event T:
T = "All the rest of the information about the other five ossuaries".
12
We are interested to know how this new information T changes our naïve
estimate P(J). In other
words, we want to know the probability that Jesus of Nazareth is
really the Jesus of the tomb,
given all the other information we have from the other ossuaries. That
is, we want P(J|T).
Bayes' Theorem then tells us that:
EURO
P(J |T) =
P(T | J)P(J)
P(T | J)P(J) + P(T |~ J)P(~ J)
Bayes' Theorem takes a bit of getting used to, but it corresponds to
Jacobovici's and our intuition.
For example, if Jesus of Nazareth were really buried in this family
tomb, we would expect
the other names in the tomb to correspond to his family members. That
is, we would expect
P(T|J) to be rather high. Whereas if Jesus of Nazareth were not buried
in the tomb, we would
expect the names in the tomb not to correspond well with his family
members. That is, we would
expect P(T|~J) to be rather low.
Bayes' Theorem tells us how to weigh the two cases against each other
to make the best decision
we can in light of the information we have. If new information were to
come to light, that would
change the calculation.
We have already shown in principle how to compute P(J) - the initial
naïve estimate that Jesus
of Nazareth is in the tomb. From that, we immediately get P(~J). So it
remains only to compute
the two conditional probabilities P(T|J) and P(T|~J). These are not
hard, and we will spend the
bulk of this article estimating these.
First, however, there are a few preliminary issues to work through.
Preliminaries to P(T|J) and P(T|~J)
First, we will deal with the issue of the ossuary "Judah son of
Jesus." If the Jesus of the tomb
were some randomly chosen man on the street, he would have a high
probability of having a son.
Most Jewish men of the period were married and did their best to obey
the commandment to be
13
fruitful and multiply. We do not know the exact probability that any
given man would have a
son, but it is likely to be near 1.
That is not the case for Jesus of Nazareth. In all of history, nobody
seems to have ever postulated
that Jesus had a son. Dan Brown, in the DaVinci Code, followed a long
and hazy tradition
that postulates a daughter. Did Jesus of Nazareth have a son? We do
not know for sure. What
we can say is that he might conceivably have had a son but that he is
not more likely than a randomly
chosen man of Jerusalem to have had a son. So we can define another
"fuzzy factor" here
to incorporate this range of possibilities. Define
EURO
F2 as follows:
EURO
F2 = the relative probability that Jesus of Nazareth had a son
By "relative probability" we mean the ratio of the probability that
Jesus had a son to the probability
that any other man of Jerusalem had a son. We expect
EURO
F2 to be between 0 and 1. Different
historians will assign different values to
EURO
F2. We will see at the end of this calculation
whether it makes much difference what value you assign to
EURO
F2.
We also define
EURO
Pson-Judah to be the absolute probability that a randomly chosen man
of Jerusalem
would have had a son named Judah.
Then, by definition, the probability that Jesus of Nazareth had a son
named Judah is
EURO
F2Pson-Judah .
We will never actually need to compute
EURO
Pson-Judah . It will factor out of our equations completely,
leaving us with only the fuzzy factor
EURO
F2, which is nicely bounded between 0 and 1.
It is worth defining exactly what we mean by the event T. What we
really care about is the set of
all possible events like the set of names observed in the tomb. That
is, we want to consider all
sets of names that would generate the same level of surprise. We have
already dealt with the
surprises of two of the ossuaries. There remain four, containing two
women and two men.
What is surprising is that two of these had the names of members of
the immediate family of Jesus.
We found a Mary and we found a Jose. Each of those coincides with
names of known
members of the family of Jesus. We also found a second Mary, who is
known by DNA analysis
14
not to be a member of the immediate family of Jesus. We also found a
Matthew, who is not in
the known immediate family of Jesus.
If we opened a random tomb in Jerusalem and found two women and two
men, what are the odds
that at least one of the women would be a Mary and at least one of the
men would have the name
of a brother of Jesus? That is the question that P(T|~J) will answer.
Likewise, if we had opened the actual tomb of the family of Jesus and
found the names of two
women and two men, we must also ask what are the odds that at least
one of the women would
be a Mary and at least one of the men would have the name of a brother
of Jesus. Neither of
these is a sure thing, even in the family tomb of Jesus. That is the
question that P(T|J) will answer.
In order to answer these questions, we need to know the probability of
drawing the names of
close family members of Jesus if we choose people from ancient
Jerusalem at random. We need
census data, and we do not have it. But we do have two surveys of
names from Israeli scholars,
Rachel Hachlili and Tal Ilan.11 Their numbers do not agree perfectly,
but they are reasonably
close to each other.
In the following table, we summarize the relevant data. Note that
names took many forms in ancient
Israel. We consider all variant forms to be essentially the same. So
the Hebrew/
Aramaic/Greek names Miriam, Mariam, Maria, Mariame, Mariamme,
Mariamne, Mariamene,
Mariamenon, etc., all count as Mary. Likewise, the variant forms
Yosef, Yehosef, Yosi,
Yoseh, Yosah, etc., all count as Joseph.
We will also assume that the probability of finding a "Jesus son of
Joseph" is obtained by multiplying
the probability of finding a "Jesus" by the probability of finding a
"Joseph". Note that
Jesus had four brothers, James, Simon, Judah, and Jose. Jose is a
variant form of Joseph.
11 See Rachel Hachlili. "Names and Nicknames of Jews in Second Temple
Times," Eretz Israel 17, 1984, and Tal
Ilan. Lexicon of Jewish Names in Late Antiquity: Part 1, Palestine 330
BCE-200 CE. Tubingen, Germany: Mohr
Siebeck, 2002.
15
Here are the probabilities that we will need:
Name Probability (Hachlili) Probability (Ilan)
Jesus 0.09 0.0411
Joseph 0.14 0.0921
Jesus son of Joseph 0.0126 0.0038
Simon 0.21 0.1024
Judah 0.10 0.0713
James 0.02 0.0179
Joseph 0.14 0.0921
Any Brother of Jesus 0.47 0.2837
Mary 0.214 0.254
Since we have two choices for the probability distributions of names,
we will need to run our
calculations twice, once for Dr. Hachlili's numbers and once for Dr.
Ilan's numbers.
We can now estimate the number of men in Jerusalem who might be named
"Jesus son of Joseph."
Suppose there were N people in Jerusalem, and suppose the practice of
burial in ossuaries
continued for g generations. The practice went on for about 90 years,
so g will be more than 1
but less than 3. Since half the inhabitants of Jerusalem were males,
we estimate the number of
men named "Jesus son of Joseph" to be:
EURO
NJ =
1
2
NgPJesusPJoseph
Estimating N = 80,000, g = 2, we find a value roughly between 300 and
1000.
We are now ready to compute P(T|J) and P(T|~J). The second of these is
a bit easier to compute,
so we will do it first.
Computing P(T|~J)
16
If we know for sure that the ossuary labeled "Jesus son of Joseph" did
not contain Jesus of Nazareth,
what are the odds of observing the other five ossuaries in the tomb?
That is quite easy to compute. There are three factors:
1) The probability of the "Judah son of Jesus" ossuary, which is
EURO
Pson-Judah
2) The probability of finding at least one "Mary" ossuary out of two
ossuaries containing
randomly chosen women.
3) The probability of finding at least one ossuary with the name of a
brother of Jesus,
out of two ossuaries containing randomly chosen men.
Define
EURO
PMary to be the probability of a randomly chosen woman being named
Mary. We can estimate
this using either Rachel Hachlili's data or Tal Ilan's data, given in
the table above.
Then the probability of a randomly chosen woman not being a Mary is
EURO
1- PMary. Likewise, the
probability that neither of two randomly chosen women will be a Mary
is
EURO
(1- PMary )2. Therefore,
the probability that at least one of two randomly chosen women will be
a Mary is
EURO
1- (1- PMary )2 .
Likewise, define
EURO
PBrother to be the probability of a randomly chosen man having a name
that
matches one of the brothers of Jesus of Nazareth. Again, we can
estimate this probability by
using either the Hachlili or the Ilan data. In either case, we
likewise compute the probability that
at least one of two randomly chosen men will have a name that matches
one of the brothers of
Jesus is
EURO
1- (1- PBrother )2.
Now we can put all this together to get what we are after:
EURO
P(T |~ J) = Pson-Judah[1- (1- PMary )2][1- (1- PBrother )2]
Computing P(T|J)
If we know for sure that the ossuary labeled "Jesus son of Joseph" did
contain Jesus of Nazareth,
what are the odds of observing the other five ossuaries in the tomb?
This is again fairly easy to
compute. As before, there are three factors:
1) The probability of the "Judah son of Jesus" ossuary, which is
EURO
F2Pson-Judah
17
2) The probability of getting at least one "Mary" ossuary out of two
ossuaries chosen
randomly from women in the extended family of Jesus.
3) The probability of getting at least one "brother" ossuary out of
two ossuaries chosen
randomly from men in the extended family of Jesus.
A new wrinkle comes in here. Since this is definitely the family tomb
of Jesus we are considering,
we need to make sure that the family of Jesus is actually in it. This
means that we should
ensure from the beginning that the tomb contains:
1) Mary, the mother of Jesus, plus a number of other unknown women
2) James, Simon, Joseph, and Judah, the four brothers of Jesus, plus a
number of other
unknown men
This should be obvious. When we say "this is the family tomb of
Jesus," what we mean is that
his immediate family is in it, plus his (unknown) extended family. The
intuition here is that all
immediate family members, except the father Joseph, would be expected
to be buried here.
However, as is the case with all family tombs - some family members
receive inscriptions while
others do not. Thus, our interest here is in estimating the
probability that the known family
members of Jesus of Nazareth's family would receive inscriptions in a
family of size 2n+2.12
Assuming that the extended family members have names that are
distributed according to Ilan or
Hachili, let us work out the consequences of all this. Besides "Jesus
son of Joseph" and "Judah
son of Jesus," we will assume that the tomb contains n men and n
women, but the distribution of
names will be slightly distorted, because we have stipulated a few
names. We can work out the
new distribution of names quite easily.
The expected number of women named Mary is now:
EURO
E(Mary) =1+ (n -1)PMary = nPMary + (1- PMary )
12 Implicit in this is the assumption that a named member of Jesus of
Nazareth's family is just as likely as a nonnamed
member of the family to receive an inscription. This assumption may or
may not correspond with reality.
However, the extent to which it does not would only be insofar as we
would expect the named family members to be
more likely to receive an inscription. If that is the case - if, say,
Simon is more likely than this "Matthew" to receive
an inscription - the probability that this is Jesus of Nazareth's tomb
will drop because it becomes more and
more unlikely that only 2 members of the known family would have an
inscription. Thus, this assumption - insofar
as it fails to correspond to reality - is favorable to Jacobovici's
thesis. This is important to note. We are offering a
critique of the hypothesis, but the critique itself is not predicated
on any assumption we have made. On the contrary,
our assumptions, where and when we are forced to make them, favor
Jacobovici's thesis and thus do not diminish
the analytical power of our critique.
18
This implies that the new probability for finding a Mary within this
family tomb of n women is:
EURO
PMary
* (n) = PMary +
(1- PMary )
n
This confirms what we intuitively expect. Within the family tomb of
Jesus, it is more likely to
find a woman named Mary. But the "Mary-enrichment" effect decreases as
the number of
women in the tomb increases.
Likewise, the expected number of men having names of the brothers of
Jesus is
EURO
E(Brothers'names) = 4 + (n - 4)PBrother = nPBrother + 4(1- PBrother )
This implies that the new probability for finding a man with the name
of a brother of Jesus
within this family tomb of n men is:
EURO
PBrother
* (n) = PBrother +
4(1- PBrother )
n
Again, this is intuitively clear. Within the family tomb of Jesus, it
is more likely to find a man
with the name of one of his brothers, but this "brother-enrichment
effect" decreases as the number
of men in the tomb increases.
We can now put all this together to compute the term we wanted all
along:
EURO
P(T | J) = F2Pson-Judah[1- (1- PMary
* )2][1- (1- PBrothers
* )2 ]
At this point, it is natural to ask about the practice of naming sons
after fathers. Isn't it more
likely to have a son named Joseph in a family that already has a
Joseph as the father?
The answer is yes, it does appear that families did quite often name
one of their sons after the
father. This is certainly true in the family of Jesus - he had a
brother Joseph who was named
after his father. The New Testament calls this brother both
"Joseph" (Matthew 13) and "Joses"
(Mark 6).
19
Will this effect change our calculation somehow? Will it enrich the
number of expected Josephs
in the family tomb of Jesus?
The answer is no, it will not. We already knew that Jesus had a
brother named Joseph, so we
have already enforced that fact above. So there is no "tendency" to
account for in the family of
Jesus. We enforced it already as a fact.
However, this issue does suggest that we should revisit our previous
calculation for P(T|~J) and
account for the "Joseph enrichment factor" there. We will turn to that
now.
Revised Computation of P(T|~J)
If we know that our tomb does not contain the family of Jesus of
Nazareth, we need to account
for the "Joseph enrichment factor" which will increase the probability
of finding a son of Joseph
named after him in this tomb. But there are actually two effects to
account for.
In a family tomb with a "Jesus son of Joseph," there is a fairly high
probability that the father
Joseph of this Jesus will also be buried in the tomb. We need to
account for that, too.
It may be, of course, that we are barking up a wrong tree and that
neither of these really matter.
In that case, there will be no increase in the expected number of
Josephs in this family tomb.
That is one extreme.
On the other hand, it may be that we should always expect to find both
the father Joseph and a
son named after him, in which case we would find two more Josephs in
such a tomb than in an
ordinary one.
Neither of these is a likely scenario. The answer is probably
somewhere between these extremes.
20
Let us define a new "fuzzy factor"
EURO
F3 to account for our lack of knowledge.
EURO
F3 will be a number
between 0 and 2 that represents the increase in the expectation value
of the number of Josephs in
a family tomb that contains the name "Jesus son of Joseph".
Since the expectation value in a tomb with n men was already
EURO
nPJoseph , it should be clear that the
enhanced probability of finding a Joseph in such a tomb is exactly:
EURO
PJoseph
+ = PJoseph +
F3
n
When
EURO
F3 is set to 0, there is no enhancement at all. That is the value we
should choose in societies
where families are neither more nor less likely to name one son after
the father and where the
father of a man named "Jesus son of Joseph" is never buried in the
family tomb with him.
EURO
F3 should be set to 2 in societies where the probability of naming a
son after the father is 1 and
where the probability of a father being buried with his son is also 1.
We expect that
EURO
F3 should be somewhere between 0 and 2, but we do not really know
where it
should be for the society of ancient Jerusalem. When we run
calculations, we will be interested
to know whether
EURO
F3 makes any difference or not, so we will run calculations over a
range of values
for
EURO
F3.
There is a corresponding enrichment in the probability that a randomly
drawn man in the tomb
will have one of the names of the brothers of Jesus of Nazareth. This
probability is now:
EURO
PBrother
+ (n) = PBrother +
F3
n
The net effect of all this is to change our previous estimate of P(T|
~J) slightly:
EURO
P(T |~ J) = Pson-Judah[1- (1- PMary )2][1- (1- PBrother
+ )2]
Final Computation
21
Now we can put all our results together. Recall that we are trying to
compute:
EURO
P(J |T) =
P(T | J)P(J)
P(T | J)P(J) + P(T |~ J)P(~ J)
We have estimated the following:
EURO
P(J) =
F1
NJ
P(~ J) =1- P(J)
EURO
P(T | J) = F2Pson-Judah[1- (1- PMary
* )2][1- (1- PBrother
* )2]
EURO
P(T |~ J) = Pson-Judah[1- (1- PMary )2][1- (1- PBrother
+ )2]
where:
EURO
NJ =
1
2
NgPJesusPJoseph
EURO
PMary
* (n) = PMary +
(1- PMary )
n
EURO
PBrother
* (n) = PBrother +
4(1- PBrother )
n
EURO
PBrother
+ (n) = PBrother +
F3
n
Note that the completely unknown probability
EURO
Pson-Judah factors out of the calculation, since it is a
factor in both the numerator and denominator of the Bayes' Theorem
equation.
The resulting equation is computable as a function of n and the
various fuzzy factors
EURO
F1,
EURO
F2,
and
EURO
F3. The first two of these fuzzy factors are required to be between 0
and 1, and the last is
required to be between 0 and 2 so the fuzziness is nicely bounded.
22
Results
In this section, we will compute the results of P(J|T) for a variety
of choices of the input parameters.
Again, the final results will vary depending upon the choices made
with regard to the
fuzzy factors we have delineated. Rather than provide our own estimate
of the probability,
which would require us to engage in a historical/archaeological
discussion that is beyond our
purview, we choose rather to offer five estimates that differ in the
choices of the fuzzy factors
and, accordingly, P(J|T). We also choose the most and least optimistic
values for each fuzzy
factor, which as we have discussed will vary individually over a
bounded range, so as to provide
a bounded range of potential values that P(J|T) might take.
Readers who would like to check our calculations (or run new ones with
different assumptions)
can download our spreadsheet from the following page:
http://www.ingermanson.com/jesus/art/stats2.php
The spreadsheet allows you to set six variables:
(a) The number of people in Jerusalem
(b) The number of complete generations during which ossuaries were in
use
(c) Use Rachel Hachlili's population statistics or use Tal Ilan's
(d) F1: The relative probability that Jesus had a son
(e) F2: The relative probability that Jesus was interred in a tomb
like the one at Talpiot
(f) F3: The number of "extra Josephs" one would expect in a family
tomb containing a
man named "Jesus son of Joseph"
23
Case 1: Advocate who wants the tomb to belong to Jesus of Nazareth
The numbers are chosen to be as favorable as possible to the "tomb
hypothesis." This is the case
no matter how absurd the assumptions might be from a historical
perspective. The intention here
is to make P(T|J) as high as possible, regardless of rationality:
(a) N = 30,000 (Number of inhabitants of Jerusalem. Set as low as
possible.)
(b) g = 1 (Number of generations in Jerusalem using ossuaries. Set as
low as
possible.)
(c) Tal Ilan's probabilities for names
(d) F1 = 1.0 (Relative probability that Jesus had a son. Jesus was
just as likely as
any man to have a son.)
(e) F2 = 1.0 (Relative probability that Jesus could be buried in
Talpiot. Jesus was
just as likely as any man to be buried in Talpiot.)
(f) F3 = 0.0 (Expected number of extra Josephs in a tomb with a "Jesus
son of Joseph"
ossuary. No extra men named "Joseph" are to be expected.)
The results are plotted for a number of bodies in the tomb ranging
from 10 up to 36:
P(J|T) vs. Number of bodies in tomb
0
0.01
0.02
0.03
0.04
0.05
0.06
0 10 20 30 40
P(J|T)
P(J|T)
Conclusion: The probability that the tomb belongs to Jesus of Nazareth
could conceivably be as
high as 1 in 18 if all factors are made as favorable as possible,
irrespective of historical plausibility.
24
Case 2: "Indifferent" historian
The numbers here are chosen with the concept of indifference in mind.
That is, the historian in
this instance does not have a relatively strong opinion on any of the
fuzzy factors. This is not to
be taken as our estimation of the average scholarly response, which is
probably better modeled as
being somewhere between this and Case 4.
(a) N = 50,000 (Number of inhabitants of Jerusalem)
(b) g = 2 (Number of generations in Jerusalem using ossuaries)
(c) Tal Ilan's probabilities for names
(d) F1 = 0.01 (Relative probability that Jesus had a son)
(e) F2 = 0.5 (Relative probability that Jesus could be buried in
Talpiot)
(f) F3 = 1.0 (Expected number of extra Josephs in a tomb with a "Jesus
son of Joseph"
ossuary)
The results are plotted for a number of bodies in the tomb ranging
from 10 up to 36:
P(J|T) vs. Number of bodies in tomb
0
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
0 10 20 30 40
P(J|T)
P(J|T)
Conclusion: The probability that the tomb belongs to Jesus of Nazareth
is less than 1 in 19,000,
with a choice of parameters likely to be made by a historian
indifferent to the outcome.
25
Case 3: Historian inclined toward theory
The numbers are chosen to be quite favorable to the "tomb hypothesis,"
but staying within the
bounds of historical reasonableness:
(a) N = 50,000 (Number of inhabitants of Jerusalem)
(b) g = 2 (Number of generations in Jerusalem using ossuaries)
(c) Tal Ilan's probabilities for names
(d) F1 = 0.1 (Relative probability that Jesus had a son)
(e) F2 = 0.8 (Relative probability that Jesus could be buried in
Talpiot)
(f) F3 = 1.0 (Expected number of extra Josephs in a tomb with a "Jesus
son of Joseph"
ossuary)
The results are plotted for a number of bodies in the tomb ranging
from 10 up to 36:
P(J|T) vs. Number of bodies in tomb
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
0 10 20 30 40
P(J|T)
P(J|T)
Conclusion: The probability that the tomb belongs to Jesus of Nazareth
is less than 1 in 1,100,
even when choosing relatively liberal values of parameters so as to
favor the tomb hypothesis.
26
Case 4: Historian disinclined toward theory
The numbers are chosen to be unfavorable to the "tomb hypothesis"
while still being defensible
from a historical perspective. Based upon an informal survey of most
responses to the documentary's
hypothesis - it seems that many scholars would place themselves here.
That is, they
are inclined to assign low values to the "fuzzy factors":
(a) N = 80,000 (Number of inhabitants of Jerusalem)
(b) g = 2 (Number of generations in Jerusalem using ossuaries)
(c) Rachel Hachlili's probabilities for names
(d) F1 = 0.001 (Relative probability that Jesus had a son.)
(e) F2 = 0.01 (Relative probability that Jesus could be buried in
Talpiot)
(f) F3 = 1.5 (Expected number of extra Josephs in a tomb with a "Jesus
son of Joseph"
ossuary)
The results are plotted for a number of bodies in the tomb ranging
from 10 up to 36:
P(J|T) vs. Number of bodies in tomb
0
0.000000002
0.000000004
0.000000006
0.000000008
0.00000001
0.000000012
0.000000014
0.000000016
0.000000018
0.00000002
0 10 20 30 40
P(J|T)
P(J|T)
Conclusion: The probability that the tomb belongs to Jesus of Nazareth
is less than 1 in 5 million
if all factors are made unfavorable.
27
Case 5: Christian who insists that Jesus' body ascended to heaven
Only one number matters:
(e) F2 = 0.0 (Relative probability that Jesus could be buried near
Talpiot)
The results are plotted for a number of bodies in the tomb ranging
from 10 up to 36:
P(J|T) vs. Number of bodies in tomb
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40
P(J|T)
P(J|T)
Conclusion: If Jesus' body ascended to heaven, then the probability
that this is his tomb is 0.
Assigning a Range to P(J|T)
Finally, it is worthwhile to assess the range of values that P(J|T)
might take. Unfortunately, we
have six factors that vary, which makes graphical representation
impossible. Thus, let us vary
two factors, F1 and F2, simultaneously. Recall that F1 is the relative
probability that Jesus of
Nazareth had a child and F2 is the relative probability that Jesus of
Nazareth could be buried at
Talpiot.
The remaining factors will be held constant according to the
assumptions in the "indifferent historian"
case. Thus, N = 50,000, g = 2, F3 = 1 and Tal Ilan's probabilities are
used. With these
conditions, we obtain the following results:
28
F2
0.001 0.01 0.9 1.0
F1 0.001 0.0000000105 0.0000002092 0.0000094574 0.0000105138
0.01 0.0000001046 0.0000010459 0.0000945656 0.0001051276
0.9 0.0000094124 0.0000941210 0.0084398749 0.0093737827
1.0 0.0000104583 0.0001045778 0.0093688530 0.0104044776
The rows represent the different values for F1, the columns for F2 and
the cells are the calculations
of P(J|T) for every given F1 and F2. Two observations stand out
immediately. First, while
it is the case that varying F1 and F2 have a large relative effect -
that is varying from the lowest
F1/F2 to the highest amounts to a 1 million-fold increase - the
absolute effect is minimal. That
is, varying the factors from the least optimistic to the most
optimistic only "purchases" a single
percentage point. Second, the numbers are consistently low. Regardless
of the value of either F1
or F2, the likelihood that this tomb belongs to Jesus of Nazareth is
consistently small. Even assuming
that he is as likely as any man to have a son and to be buried at
Talpiot - one still can
find only a 1.04% probability that the Talpiot tomb belongs to him.
Discussion
Our analysis thus points in a very different direction than the
conclusion of Jacobovici. Why is
this the case? The answer to this question can best be seen by
computing a simple estimate of
P(T|J). Let us, in this instance, remove all of the fuzzy factors. In
other words, let us say that
Jesus of Nazareth is just as likely as any man to have had a son (i.e.
F1 = 1) and just as likely as
any man to have been interned at Talpiot (i.e. F2 = 1). Let us also
say that we expect no extra
Josephs in the tomb (i.e. F3 = 0). Additionally, let us take the rest
of the assumptions made by
our "indifferent historian" in Case 2 above. Thus, we assume a
population of Jerusalem of
50,000 people and the total number of generations interned in
ossuaries is two. We also use Tal
Ilan's demographic data. Finally, we assume only 10 people were
interned in the tomb.13
13 All of these have the effect of maximizing the value of P(J|T)
while retaining reasonable assumptions about demography.
29
We still use our basic Bayes' Theorem formulation. That is:
EURO
P(J |T) =
P(T | J)P(J)
P(T | J)P(J) + P(T |~ J)P(~ J)
With the stated assumptions, we obtain:
P(T|J) = 0.687
P(T|~J) = 0.216
P(J) = 0.005
P(~J) = .995
Returning to our equation, we obtain:
( | ) .014 1.4%
..218
..003
( | )
0.687 0.006 0.216 .995
0.687 0.005
( | )
= =
=
× + ×
×
=
P J T
P J T
P J T
This is far from the 600 to 1 chance hypothesized by Jacobovici. The
reason should be clear
when we recall the fundamental intuition of Bayes' Theorem. We are
interested in defining
P(J|T) in terms of P(T|J). This requires us to consider two factors,
neither of which Jacobovici
considered.
The first is the unlikelihood of ever finding Jesus of Nazareth in the
first place. This value -
even under these very optimistic assumptions - is quite low, just
0.5%. Relatedly, the likelihood
that we have not found Jesus of Nazareth is a very sizeable 99.5%.
This must be considered in
any evaluation of the tomb. To fail to do so is, again, to commit what
is known as the prosecutor's
fallacy. One cannot focus solely upon the uniqueness of the evidence
gathered. Its uniqueness
must be weighed against the uniqueness of the claim. While Talpiot
might have some
unique evidence that points to Jesus of Nazareth, its descriptive
power is necessarily diminished
when we consider just how unlikely it is that we would ever find his
tomb in the first place.
30
The second is the extent to which the names in the tomb provide a clue
to which Jesus was interned
there. The good news for Jacobovici's hypothesis: we can expect with
68.7% likelihood
that Jesus of Nazareth would be buried with at least one Mary and at
least one brother. The bad
news: we can also expect 21.6% of all people named "Jesus son of
Joseph" who are not Jesus of
Nazareth to be buried similarly. In other words, Talpiot does not
provide all that unique a clue at
all. Why is this the case? It is for the reason that historians,
archaeologists and New Testament
scholars have been stating since the day the film was announced: these
names are common.
Thus, while it is true that finding four particular common names in a
cluster is uncommon, as
Jacobovici and others have responded, this is beside the point. The
point is that the names in Jesus
of Nazareth's family were not sufficiently unique such that a tomb
that matches two of them
decisively points toward Jesus of Nazareth. Indeed, the difference
between P(J|T) and P(T|~J) is
far from decisive. Many men named "Jesus son of Joseph" can be
expected to have been buried
with at least one woman named Mary and at least one man with a name
that follows the names of
Jesus of Nazareth's brothers. Common names mean that the names
themselves do not take us
very far in terms of identifying the owner of the tomb.
Thus, it should be clear that Jacobovici's inferential errors had a
sizeable and beneficial effect on
his hypothesis. The failure to consider P(J) and P(T|J) dramatically
inflated the estimate of
P(J|T), biasing it (in the statistical sense of the term) from its
expected value and toward the conclusion
of the documentary. Factoring in the miscellaneous computational
errors we also reviewed,
it should be clear why our figure is dramatically lower.
When we begin to consider the fuzzy factors, the value of P(J|T)
begins to drop even further. It
is important to note that it can only fall from this value of 1.4%.
Jacobovici implicitly assumed
F1 = 1, F2 = 1, and F3 = 0. All are unrealistic assumptions. If we
adjust them to be more realistic,
P(J|T) will fall below 1.4%. Much of the "damage" will be done by F1,
the fuzzy factor that
assigns a relative probability to Jesus of Nazareth having a child.
Given the unanimous and
deafening silence of the historical record, it seems to us unlikely
that this value could be taken
above 0.05 without resort to a tendentious and error-filled argument.
With F1 = .05, F2 = 1 and
F3 = 0, the value of P(J|T) drops to 0.08%. When F2 is lowered and F3
is raised to correspond
better with our expectation of reality, P(J|T) will drop even further.
31
Conclusion: Statistics above All?
Rather than conclude by recapitulating our hypothesis, we feel we
should take the opportunity to
respond to an insightful argument offered by Professor James Tabor,
who has been a thoughtful
supporter of the theory of Jacobovici. On his Jesus Dynasty Blog,
Tabor writes:
A statistician, as statistician, is not primarily focusing on
prosopography, that is,
matching ancient names to known historical characters. That is the
task of the
historian who then seeks to determine if there is any potential "fit"
between this
cluster of names, with its configurations, and that of any
identifiable persons/
family in our records...
I am not optimistic that more advanced statistical models can be
effectively applied
to questions of historical prosopography since the kinds of
identifications
and subtle correspondences used are not easily quantified. Is
Mariamene an appropriate
name for [Mary Magdalene]? How could you put a number on it? Is it
significant that her ossuary is decorated and her inscription is in
informal Greek?
How is that quantified? Does it matter that the name Yeshua bar
Yehosef is
written in a very messy graffiti style while the others are elegant
and block? How
do you put a number on that? What of how the ossuaries were placed in
the various
kokim, and with names grouped in twos and threes? Are there hints of
potential
relationships implied? I have about 25 other factors of this sort that
I am considering
in formulating my own prosopographic proposal, including the symbol
on the tomb that comes from contemporary temple gate imagery. As far
as I can
tell many of these factors can not be quantified.14
This raises an interesting question: does statistical analysis such as
the kind that we have outlined
here preclude historical and archaeological analysis?
While we find ourselves agreeing with Dr. Tabor on several important
points, we believe that the
answer is no. To clarify our position, it is valuable to outline what
we believe we have and have
not delineated in this article.
At a fundamental level, statistical analysis is a formalization of the
proper manner of descriptive
inference. Thus, we would argue that the first accomplishment we have
made in this essay is
formally to outline how analysis of the Talpiot tomb should proceed.
This is the case even if one
14 James Tabor. "Probabilities, Statistical Theory, and the Talpiot
Tomb." March 19, 2007.
http://jesusdynasty.com/blog/2007/03/19/probabilities-statistical-theory-and-the-talpiot-tomb/.
Accessed March 26,
2007.
32
ultimately decides that a formal probability is incalculable. Even if
it is impossible to assign actual
values to the sundry variables that we have delineated - one
nevertheless must take these
variables into account. This is, as we indicated above, the
fundamental methodological mistake
of Jacobovici. He failed to consider P(J) and P(T|J). Even if he had
chosen a non-quantitative
route for his argument, the failure to consider these factors would
still invalidate his conclusions.
The reason is that, even if one chooses to avoid the mathematics
behind statistical analysis, one
must still obey the rules of descriptive inference it delineates. When
one fails to do this, one runs
the risk of committing an inferential fallacy of some sort - just as
Jacobovici has committed the
prosecutor's fallacy.15
In addition to outlining the proper manner in which an argument about
Talpiot should proceed,
we also would argue that we have here delineated a range of values
that the final P(J|T) can
take. In other words, we are in full agreement with Dr. Tabor that the
job of the "statistician"
does not preclude the job of the historian or the archaeologist. This
is why we have delineated a
series of "fuzzy factors" and have demurred from assigning final
values to them. We are not
formally trained in either history or archaeology. It is not our
business to assign values to these
numbers. What we have done, however, is identify where the historical
and archaeological debates
can influence the final outcome. While we might lack the scholarly
background in history
to predict the results of the sundry debates that are ongoing, we
nevertheless have the capacity to
identify where, how and why those results will affect the final
estimate of P(J|T). For instance, a
prosopographical discussion of the family of Jesus of Nazareth, above
all a serious investigation
of the likelihood that he had a son named "Judah," will influence P(T|
J). An archaeological/
historical discussion about whether one might expect Jesus of Nazareth
to be interned in a
15 Dr. Tabor, in arguing for the more simple method advanced by Dr.
Feuerverger and explicitly against our method
(which he calls "a more Bayesian model"), asserts that the measure of
Feuerverger embodies what he calls "the
Ockham's razor of probability theory." As it is, however, any argument
about the ownership of this tomb that relies
upon Feuerverger's figure without reference to the other factors we
have delineated will be guilty of the inferential
errors we have also delineated. And while, following Ockham,
simplicity is a virtue in any argument, it is not the
highest virtue: simple arguments with inferential errors are not to be
preferred to (slightly) more complicated arguments
that lack such errors. We would also note that while we discussed our
basic hypothesis with Dr. Tabor, we
did not give him an advance copy of this essay. Accordingly, we are
not sure that his knowledge of our methodology
was sufficient for this critique, which predates the public release of
this essay. See James Tabor, "Statistical
Clouds, Fuzziness, and Ockham's Razor." March 25, 2007.
http://jesusdynasty.com/blog/2007/03/25/statisticalclouds-
and-fuzziness. Accessed March 26, 2007.
33
tomb like the Talpiot tomb, placed in the particular ossuary in
question, and so on, will influence
P(J).
These debates are therefore embodied in our "fuzzy factors," which
each vary over a small
range. While we cannot predict what their particular values will be,
and while we recognize
(following Dr. Tabor) that it is possibly the case that their expected
values cannot be quantified
because it would require placing numbers on what are essentially non-
numerical concepts, we
can nevertheless vary these factors between 0 and 1 to see how this
variation affects P(J|T). It is
in this manner that we can offer a range of values for P(J|T). This is
what we have done.
What we have not done is delineate an expected value for P(T|J), which
we would agree with Dr.
Tabor might in fact be incalculable. This would require us to take
positions on matters which are
beyond our purview, namely the historical/archaeological debate that
is now ongoing, and to
quantify what is possibly not quantifiable.
Thus, we find ourselves in both agreement and disagreement with Dr.
Tabor. On the one hand,
we think that statistical analysis cannot stand as a substitute for
the historical/archaeological debate
- and that, at some point, certain unquantifiable elements might
preclude the assignment of
an expected value. On the other hand, we think that statistical
analysis can be taken further than
Dr. Feuerverger has taken it, and further than Dr. Tabor believes it
can be taken.
Specifically, we think that our model possesses the right combination
of flexibility and clarity
not only to allow for the prosopographical debate Dr. Tabor rightly
wishes to see commence, but
also to indicate where the results of that debate will have an effect
and what kind of effect that
might be. Generally, we believe that it can offer two valuable
services: (a) the methodological
outline of how any argument - quantitative or non-quantitative -
should proceed; (b) a range of
values for P(T|J).
34
About the Authors
Jay Cost is a doctoral candidate in political science at the
University of Chicago. His scholarly interests include
contemporary American political parties, congressional elections, and
the media's role in politics. He is also a
regular contributor for RealClearPolitics.com, a Chicago-based
organization that provides a daily digest of the best
commentary and analysis of American politics.
Randy Ingermanson earned his Ph.D. in physics from the University of
California at Berkeley. He has worked for
many years as a computational physicist. Randy is the author of six
award-winning novels, and he also authored a
nonfiction book, Who Wrote the Bible Code?, which applied statistical
methods to debunk the alleged "Bible code."
He is currently working on a novel about Jesus. His web site is at
http://www.Ingermanson.com.
.