A little statistics and the tomb of Jesus
Have you been following the story of the Tomb of Jesus? I haven't been following it in detail, I'll admit, just reading the odd blog post here and there. What I'm finding, however, is that a lot is being read into a rather crude back-of-the-envelope calculation, one which I'm fairly certain is wrong. Let's put this in perspective.
In a tomb discovered near Jerusalem, there are a number of ossuaries (bone boxes), inscribed with names. This is nothing unusual. Something on the order of a thousand tombs have been discovered in the area. The archeology based on them tells us a lot about the naming conventions in 1st century Judea. This particular tomb, however, contains ossuaries inscribed with the names Yeshua bar Yehosef (Jesus son of Joseph), Yose (an abbreviated form of Joseph), Maria (Mary), and Mariamne (an odd form of Mary). There were also the names Matia (Matthew) and Judah the son of Jesus. Now, all of these were very common names in 1st century Judea. So common, in fact, that 1 out of every 190 men were named Jesus the son of Joseph. Now James Cameron has produced a documentary, arguing that although the names were common, the odds of finding this particular combination of names in one tomb are so miniscule, that this must be the family tomb of the Jesus in the gospels. To support his argument, there's the aforementioned "back-of-the-envelope calculation." With a blog called "Back of the Envelope," I am, as you'd imagine, all for back-of-the-envelope calculations, especially the crude kind. However, I do think that if you're a world-class statistician putting your reputation on the line for a high-publicity documentary purporting to disprove the world's largest religion, you ought to maybe put a mite more effort into it. So let's take a look at the calculation involved, conveniently transcribed from the documentary's flash website by StatGuy:
There are a number of problems with this.
First, much is made that Mariamne is a distinctive form of Mary referring to Mary Magdalene, and that Mary Magdalene was the wife of Jesus. Unfortunately for the documentary, this particular belief in the marriage of Jesus and Mary Magdalene has no ancient pedigree. It can't be traced back to the gospels, or any of the early Christian writings or traditions. Not even the Gnostic gospels, such as the third century Gospel of Phillip, which is often cited by supporters of this belief, specifically make that claim. In fact, it seems to be an entirely modern invention dating back to the pseudohistorical book "The Holy Blood and the Holy Grail." It's also highly unlikely that Mariamne is, in fact, a distinctive name for Mary Magdalene. The gospels always use Maria or Mariam, and the use of Mariamne, as a variant of the Hellenized form Mariamme, doesn't appear until the Gnostic gospels in the late second century, and even there it doesn't appear to be a unique name for Mary Magdalene. Further, the name on the tomb is not Mariamne, but Mariamenou, which Richard Bauckham convincingly argues has a very different etymology.
The second may be a simple misunderstanding on my part, but several references point out that Yose is an abbreviated form of Joseph. If that is the case, is there good reason to believe that the Joseph that Jesus is the son of is not Yose? If that’s the case, the presence of Yose is most definitely not an independent variable, and should not be included in the probability calculation. Finding Joseph, Jesus the son of Joseph, and Judah the son of Jesus all in the same family tomb really only gives us one independent variable, Jesus son of Joseph, to connect to the Jesus of the Bible. Now, it is true that Jesus's brother was referred to as Yose in the gospels, so it may be argued that it is unlikely that Yose and Joseph are the same person. However, they are still related names, and having a Yose and a Joseph in the same family are still not independent probabilities.
That, however, is archeology, genealogy, and etymology, and I promised you a little statistics. In this analysis, I'll continue to include the names Yose and Mariamne, despite the aforementioned problems with them. You'll have to bear with me, as I explain the problems with the back of the envelope calculation above, but the basic problem is that the expert, Dr. Feuerverger, treats as a permutation what should really be a combination. Okay, those terms are not entirely mathematically precise here, but let me try to explain:
If you read the hidden explanation (hidden mainly because it's pretty long), I think you see where this is going. If not, well, the bottom line is that Dr. Feuerverger's calculation gives the correct result for computing the odds that a man named Jesus son of Joseph has a mother named Maria, a brother named Yose, and a wife named Mariamne, respectively, whereas the archaeological find doesn't indicate what their relationship is. The proper way of formulating the question is if four people, at random, are buried together, what are the odds that their names would be Jesus son of Joseph, Maria, Yose, and Mariamne, which is the odds calculated by Dr. Feuerverger (1/2,400,000), divided by 2^4, and multiplied by 24 (4*3*2). Wait a moment, you say. Where does the 2^4 come from? That's a normalization, assuming that there's a 50% chance the person is male and a 50% chance that the person is female. In this formulation, you only improve the odds slightly, to (1/1,600,000). Ah, but I'm not done yet. You see, there were more than four ossuaries there: there were at least six, probably ten, and possibly as many as thirty-five. Now the odds start to look better. With six, the odds of finding this combination of names is 1/107,000, with 10, it's 1/7,600, and with 35, it's 1/31. Updated: However, the approximation I'm using (that there are no repeated names requiring the removal of duplicates) begins to break down as the number of names get higher (and it becomes pretty unlikely for there to be no more than one Mary). It's accurate with 10, but doesn't work with 35. See the update below for another method, which does work. End of update
Now, if you want to prove something, show me that these people are connected in the way proposed by this presentation. Then, the odds start to work out the way they suggest.
Update: Okay, I've corrected a math error in the work above. The 1/31 probability is just too high. It's a result of my assuming that I wouldn't have to account for duplicate names, when one name, particularly Maria, is just too common for that to work. When the number of names from the tomb is small, it works fine, but as it gets bigger, so do the odds of another Maria in the works, thus creating duplicate permutations that need to be removed. You can see this by the fact that if the number of bodies increases, say to a thousand, the probability of the tomb having all four names becomes greater than one, which is impossible.
So, let's approach this a different way. This is also approximate, but this approximation becomes more accurate as the number of names becomes larger, rather than smaller. Let's start with ten Jewish names selected at random. What is the probability that there is at least one Maria among them? Hmm, that's a lot of permutations to list... but, you can find the probability that there are no Marias pretty easily. The probability of a random individual being named Maria is 1/8, so the probability of an individual not being named Maria is 7/8. Now, the probability that ten such individuals have no Marias among them is (7/8)^10. So, the odds that at least one is named Maria is 1-(7/8)^10.
So let's list them:
At least 1 Maria: 3/4
At least 1 Mariamne: 1/32
At least 1 Jesus son of Joseph: 1/38
At least 1 Yose: 2/9
So, for ten names, you get a 1/7,600 chance of a tomb having all four of the significant names, which is the same result we got with the other technique. However, if you have thirty-five names on your ossuaries, you get a different set of odds:
At least 1 Maria: 99/100
At least 1 Mariamne: 1/10
At least 1 Jesus son of Joseph: 1/11
At least 1 Yose: 3/5
This gives a tomb with thirty-five names a 1/188 chance of having all four significant names. So there's a difference of a factor of six from the other method.
I'll finish up by noting that although both techniques are correct, they're both approximate. (What do you expect from a blog named "Back of the Envelope"?) The second technique works better for a larger set of names, while the first works better for a smaller set of names. They give similar results around ten.
In a tomb discovered near Jerusalem, there are a number of ossuaries (bone boxes), inscribed with names. This is nothing unusual. Something on the order of a thousand tombs have been discovered in the area. The archeology based on them tells us a lot about the naming conventions in 1st century Judea. This particular tomb, however, contains ossuaries inscribed with the names Yeshua bar Yehosef (Jesus son of Joseph), Yose (an abbreviated form of Joseph), Maria (Mary), and Mariamne (an odd form of Mary). There were also the names Matia (Matthew) and Judah the son of Jesus. Now, all of these were very common names in 1st century Judea. So common, in fact, that 1 out of every 190 men were named Jesus the son of Joseph. Now James Cameron has produced a documentary, arguing that although the names were common, the odds of finding this particular combination of names in one tomb are so miniscule, that this must be the family tomb of the Jesus in the gospels. To support his argument, there's the aforementioned "back-of-the-envelope calculation." With a blog called "Back of the Envelope," I am, as you'd imagine, all for back-of-the-envelope calculations, especially the crude kind. However, I do think that if you're a world-class statistician putting your reputation on the line for a high-publicity documentary purporting to disprove the world's largest religion, you ought to maybe put a mite more effort into it. So let's take a look at the calculation involved, conveniently transcribed from the documentary's flash website by StatGuy:
Click on “Enter the Tomb”, immediately above tomb photo. When the next page has loaded, click on “Supporting Evidence” at the bottom right of the main window. When the next page has loaded, click on “Statistical Evidence”, the fourth item in the list to the left of the main text.
This is the full text:
Statistics Overview
Dr Andrey Feuerverger, Professor of statistics & mathematics at the University of Toronto, has concluded A [sic] high statistical probability that the Talpiot tomb is the Jesus Family tomb.
In a study, Feuerverger examined the cluster of names in the tomb.
This involved multiplying the instances that each name appeared during that time period with the instances of every other name.
To be conservative, he then divided the number by the statistical standard of 4 (or 25%) to allow for unintentional biases in the historical sources.
He then further divided the results by 1,000 to account for all tombs that may have existed in First Century Jerusalem.
Taking into account the chances that these names would be clustered together in a family tomb, this statistical study concludes that the odds — on the most conservative basis — are 600 to 1 in favor of this being the JESUS FAMILY TOMB. A statistical probability of 600 to 1 means that this conclusion works 599 times out of 600.
Statistics Tables
Frequency of names:
Jesus Son of Joseph: 1 in 190
Mariamne: 1 in 160
Matia: 1 in 40
Yose: 1 in 20
Maria: 1 in 4
Initial Computation: 1/190 x 1/160 x 1/40 x 1/20 x 1/4 = 1/97,280,000
Second Computation: Eliminating Matia since he is not explicatively [sic] mentioned in the Gospels:
1/190 x 1/160 x 1/20 x 1/4 = 1/2,400,000
Third Computation: Adjusting for unintentional biases in the historical sources:
2,400,000 / 4 = 600,000
Fourth Computation: Adjust for all possible First Century Jerusalem Tombs:
600,000 / 1,000 = 600
Probability Factor = 600 to 1
There are a number of problems with this.
First, much is made that Mariamne is a distinctive form of Mary referring to Mary Magdalene, and that Mary Magdalene was the wife of Jesus. Unfortunately for the documentary, this particular belief in the marriage of Jesus and Mary Magdalene has no ancient pedigree. It can't be traced back to the gospels, or any of the early Christian writings or traditions. Not even the Gnostic gospels, such as the third century Gospel of Phillip, which is often cited by supporters of this belief, specifically make that claim. In fact, it seems to be an entirely modern invention dating back to the pseudohistorical book "The Holy Blood and the Holy Grail." It's also highly unlikely that Mariamne is, in fact, a distinctive name for Mary Magdalene. The gospels always use Maria or Mariam, and the use of Mariamne, as a variant of the Hellenized form Mariamme, doesn't appear until the Gnostic gospels in the late second century, and even there it doesn't appear to be a unique name for Mary Magdalene. Further, the name on the tomb is not Mariamne, but Mariamenou, which Richard Bauckham convincingly argues has a very different etymology.
The second may be a simple misunderstanding on my part, but several references point out that Yose is an abbreviated form of Joseph. If that is the case, is there good reason to believe that the Joseph that Jesus is the son of is not Yose? If that’s the case, the presence of Yose is most definitely not an independent variable, and should not be included in the probability calculation. Finding Joseph, Jesus the son of Joseph, and Judah the son of Jesus all in the same family tomb really only gives us one independent variable, Jesus son of Joseph, to connect to the Jesus of the Bible. Now, it is true that Jesus's brother was referred to as Yose in the gospels, so it may be argued that it is unlikely that Yose and Joseph are the same person. However, they are still related names, and having a Yose and a Joseph in the same family are still not independent probabilities.
That, however, is archeology, genealogy, and etymology, and I promised you a little statistics. In this analysis, I'll continue to include the names Yose and Mariamne, despite the aforementioned problems with them. You'll have to bear with me, as I explain the problems with the back of the envelope calculation above, but the basic problem is that the expert, Dr. Feuerverger, treats as a permutation what should really be a combination. Okay, those terms are not entirely mathematically precise here, but let me try to explain:
If you read the hidden explanation (hidden mainly because it's pretty long), I think you see where this is going. If not, well, the bottom line is that Dr. Feuerverger's calculation gives the correct result for computing the odds that a man named Jesus son of Joseph has a mother named Maria, a brother named Yose, and a wife named Mariamne, respectively, whereas the archaeological find doesn't indicate what their relationship is. The proper way of formulating the question is if four people, at random, are buried together, what are the odds that their names would be Jesus son of Joseph, Maria, Yose, and Mariamne, which is the odds calculated by Dr. Feuerverger (1/2,400,000), divided by 2^4, and multiplied by 24 (4*3*2). Wait a moment, you say. Where does the 2^4 come from? That's a normalization, assuming that there's a 50% chance the person is male and a 50% chance that the person is female. In this formulation, you only improve the odds slightly, to (1/1,600,000). Ah, but I'm not done yet. You see, there were more than four ossuaries there: there were at least six, probably ten, and possibly as many as thirty-five. Now the odds start to look better. With six, the odds of finding this combination of names is 1/107,000, with 10, it's 1/7,600, and with 35, it's 1/31. Updated: However, the approximation I'm using (that there are no repeated names requiring the removal of duplicates) begins to break down as the number of names get higher (and it becomes pretty unlikely for there to be no more than one Mary). It's accurate with 10, but doesn't work with 35. See the update below for another method, which does work. End of update
Now, if you want to prove something, show me that these people are connected in the way proposed by this presentation. Then, the odds start to work out the way they suggest.
Update: Okay, I've corrected a math error in the work above. The 1/31 probability is just too high. It's a result of my assuming that I wouldn't have to account for duplicate names, when one name, particularly Maria, is just too common for that to work. When the number of names from the tomb is small, it works fine, but as it gets bigger, so do the odds of another Maria in the works, thus creating duplicate permutations that need to be removed. You can see this by the fact that if the number of bodies increases, say to a thousand, the probability of the tomb having all four names becomes greater than one, which is impossible.
So, let's approach this a different way. This is also approximate, but this approximation becomes more accurate as the number of names becomes larger, rather than smaller. Let's start with ten Jewish names selected at random. What is the probability that there is at least one Maria among them? Hmm, that's a lot of permutations to list... but, you can find the probability that there are no Marias pretty easily. The probability of a random individual being named Maria is 1/8, so the probability of an individual not being named Maria is 7/8. Now, the probability that ten such individuals have no Marias among them is (7/8)^10. So, the odds that at least one is named Maria is 1-(7/8)^10.
So let's list them:
At least 1 Maria: 3/4
At least 1 Mariamne: 1/32
At least 1 Jesus son of Joseph: 1/38
At least 1 Yose: 2/9
So, for ten names, you get a 1/7,600 chance of a tomb having all four of the significant names, which is the same result we got with the other technique. However, if you have thirty-five names on your ossuaries, you get a different set of odds:
At least 1 Maria: 99/100
At least 1 Mariamne: 1/10
At least 1 Jesus son of Joseph: 1/11
At least 1 Yose: 3/5
This gives a tomb with thirty-five names a 1/188 chance of having all four significant names. So there's a difference of a factor of six from the other method.
I'll finish up by noting that although both techniques are correct, they're both approximate. (What do you expect from a blog named "Back of the Envelope"?) The second technique works better for a larger set of names, while the first works better for a smaller set of names. They give similar results around ten.
A Christianity post at
11:40 PM on Sat, Mar 3, 2007
Next trick is to figure out the odds that some fiction writer writing 2000 years after the fact would guess that a man and a woman who were mentioned together in some ancient documents were were actually married to each other even though the documents never alluded to this and would be correct.