# Category Archives: bayes theorem and the synoptic problem

## Bayes’ Theorem and the Synoptic Problem – Markan Priority

Now I arrive at one of the main reasons behind sloppily (re)learning Bayes’ theorem for myself: To apply it to questions about history and historicity.

A large majority of scholars agree that Mark was written first, with Matthew and Luke being reinterpretations of Mark. There are a lot of arguments for, and a decent amount of arguments against, with some scholars favoring Matthean priority and a few favoring Lukan priority. The smallest minorities favoring Johannine priority or some heretical gospel’s priority (e.g. Marcion, Egerton, &c.). The problem is that “evidence” is a tricky subject, and there should be some method of weighing the various probabilities for and against some hypothesis in relation with other competing hypotheses.

Which is exactly what Bayes’ does.

What Is Meant By Evidence?

In order to make sense of our vague feelings of certainty, we should first try to quantify it so that we have something solid to work with:

As Tom slips on the ice puddle, his arm automatically pulls back to slap the ground. He’s been taking Jiu-Jitsu for only a month, but, already, he’s practiced falling hundreds of times. Tom’s training keeps him from getting hurt.

By contrast, Sandra is in her second year of university mathematics. She got an “A” in calculus and in several more advanced courses, and she can easily recite that “derivatives” are “rates of change”. But when she goes on her afternoon walk and stares at the local businesses, she doesn’t *see* derivatives.

For many of us, rationality is more like Sandra’s calculus than Tom’s martial arts. You may think “overconfidence” when you hear an explicit probability (“It’s 99% likely I’ll make it to Boston on Tuesday”). But when no probability is mentioned — or, worse, when you act on a belief without noticing that belief at all — your training has little impact.

[…]

The problem is that the algorithms that your brain uses to perform common-sense reasoning are not transparent to your conscious mind, which has access only to their final output. This output does not provide a numerical probability estimate, but only a rough and vague feeling of certainty. Yet in most situations, the output of your common sense is all you have

“Evidence” for a Bayesian is any fact or event that places some probabilistic weight on some hypothesis. Mathematically, the probability of some hypothesis given some evidence has to be greater than the probability before looking at that evidence to count as evidence. Or, P(H | E) > P(H). So if your hypothesis only has a 50% chance of being true, evidence would be any fact or observation that makes the probability of that hypothesis more than 50%. On the other side of that, evidence against some hypothesis would be any fact, observation, event, &c. where its absence decreases the probability. Or, P(H | ~E) < P(H). Remember, absence of evidence is in fact evidence of absence. Lastly, some fact, etc. that neither increases nor decreases the probability of some hypothesis is not considered evidence; it is probabilistically independent. Or, P(H | E) = P(H).

So for this exercise, I’m going to make a couple of starting assumptions. P(Mark) will be the probability that Mark was written first. P(Matt) that Matthew was written first. P(Luke) the probability that Luke was written first, P(John) that John was written first. P(~H) will be every other hypothesis (the priority of some other unknown first gospel). Since there are five options, with the fifth being a catchall, the prior probabilities — that is, the probabilities before looking at any evidence — I’ll assume are equal. So P(Mark) = .20, and P(Mark) = P(Matt) = P(Luke) = P(John) = P(~H), which all equals 1.00 or 100%.

Next, I’ll need to explain what I mean by the various subjective “probable” phrases. This is from from pages 93-94 in Richard Carrier’s Proving History:

Virtually Impossible = 0.0001%
Extremely Improbable = 1%
Very Improbable = 5%
Improbable = 20%
Slightly Improbable = 40%
Either Way = 50%
Slightly Probable = 60%
Probable = 80%
Very Probable = 95%
Extremely Probable = 99%
Virtually Certain = 99.9999%

So for example, if I say that the probability of Mark being the shortest gospel given that Mark was written first is “very probable”, I’m saying that P(E | Mark) = 95%. This, conversely, means that the probability of Mark not being the shortest gospel given that Mark was written first is “very improbable”, or P(~E | Mark) = 5%. Similarly, when comparing hypotheses, instead of saying it is “improbable” that Matt would be written first given that Mark is the shortest gospel, I am in effect saying that P(MSG | Matt) = 20%. And just like lining up premises to make an argument, we would line up likelihoods to see where our actual probability lies using Bayes:

What is the probability that Mark was written first given Mark being the shortest gospel? This is equal to the probability that Mark is the shortest gospel given Markan priority divided by the probability that Mark is the shortest gospel given Markan priority plus the probability that Mark is the shortest gospel given Matthew was written first. Or, some unknown probability equals “very probable” times “improbable” divided by “very probable” times “improbable” plus “improbable” times “improbable”. But all of that looks confusing unless we substitute actual numbers where our probability statements are.

Another issue is that probabilities must sum to 100%. So for example, from Thinking, Fast and Slow:

In a famous study, spouses were asked, “How large was your personal contribution to keeping the place tidy, in percentages?” They also answered similar questions about “taking out the garbage,” “initiating social engagements,” etc. Would the self-estimated contributions add up to 100%, or more, or less? As expected, the self-assessed contributions added up to more than 100%

So if we are not using actual numbers for our vague “very probable” etc. statements, then we might end up with total probabilities that add up to more than 100%. Let’s say that someone is asked what the probability is that Mark was written first, then Matt, then Luke, then John, and then some unknown gospel and they respond “very probable, improbable, improbable, very improbable, very improbable” respectively. Do these five probabilities add up to 100%? Who knows! Using numerical translations of the probability statements above, this would result in 95%, 20%, 20%, 5%, and 5%, respectively. Which adds up to 205%. Not good, unless you like being Dutch booked.

So it is a fact that Mark is the shortest gospel. In other words, I’ll only attempt to include facts as E into this little thought experiment and not many other hypotheticals.

Finally, to make the conclusion as strong as possible, I should pick a subjective probability that is a bit lower than I would normally pick. This way, if the conclusion still follows from antagonistic probabilities, then I my conclusion will be that much stronger.

Going Through The Evidence – Markan Priority

This will mainly be a crib of the content from Wikipedia to keep this series short. I’ll first go through the evidence for Markan priority.

1. Mark as the Shortest Gospel (MSG)

Like I wrote above, Mark is the shortest gospel. Given Markan priority, how likely is it that we would expect this? I wrote above as an example 95%, but I’ll be attempting to not go to such extremes, which will make any conclusion I arrive at that much stronger. So instead of 95% I’ll pick 60% (Slightly Probable).

But I also need conditional probabilities for the other hypotheses like Matthean, Lukan, and some other gospel’s priority. The other two synoptic gospels seem unlikely given that they are longer than Mark. But what if there is some other synoptic we don’t know about? So Mark being the shorter gospel doesn’t seem to have an effect on this other gospel, i.e. unknown.

Thus:

P(MSG | Matt) = 40%
P(MSG | Luke) = 40%
P(MSG | John) = 40%
P(MSG | ~H) = 50%

For example, Marcion’s gospel is shorter than Luke’s, yet most scholars think that Luke is earlier than Marcion (but that could be changing). So being shorter isn’t necessarily a home run.

So our Bayes’ formula is like so:

P(Mark | MSG) = P(MSG | Mark) * P(Mark) / [P(MSG | Mark) * P(Mark)] + [P(MSG | Matt) * P(Matt)] + [P(MSG | Luke) * P(Luke)] + [P(MSG | John) * P(John)] + [P(MSG | ~H) * P(~H)]

P(Mark | MSG) = 60% * 20% / [60% * 20%] + [40% * 20%]+ [40% * 20%] + [40% * 20%] + [50% * 20%]

P(Mark | MSG) = 12% / [12%] + [8%] + [8%] + [8%] + [10%]

P(Mark | MSG) = 12% / 46%

P(Mark | MSG) = 26.08%

Now P(Mark) has gone from 20% to 26.08%, an increase of only about 6% probability. P(Matt), P(Luke), and P(John) went down from 20% to 8%, and P(~H) moved slightly from 20% to 21%.

2. Content Only Found In Mark (COFIM)

There are very few passages in Mark that are found in neither Matthew nor Luke, which makes them all the more significant. If Mark was editing Matthew and Luke, it is hard to see why he would add so little material, if he was going to add anything at all. The choice of additions is also very strange. On the other hand, if Mark wrote first, it is often the case that Matthew and Luke would have strong motives to remove these passages.

One example is Mark 3:21, where we are told that Jesus’ own family thought he was “out of his mind”. Another is Mark 14:51-52, an obscure incident with no obvious meaning, where a man with Jesus in the Garden of Gethsemane flees naked.

Significant too is Mark 8:22-26, where Jesus heals a man in a slow process involving saliva; […] both these features make it a passage more likely to be omitted than added, implying Mark wrote first.

So it’s not just that there is content only found in Mark. There happens to be content only found in Matt, content only found in Luke, and content only found in John and even in heretical gospels. It’s the nature of this content that doesn’t make sense given either Matthean or Lukan priority. So I need another conditional statement: The probability of the content on found in Mark given Markan priority = P(COFIM | Mark) = ???. How confident are scholars that this content only makes sense given Markan priority? Conversely, how confident are scholars given Markan priority if this evidence wasn’t there? This would be P(~COFIM | Mark), or the probability of not having this particular content only found in Mark given Markan priority.

Both questions are compliments of each other: P(COFIM | Makr) + P(~COFIM | Mark) = 100%. Again, a safe bet would seem to be 60% in favor, and 40% in disfavor.

A different question is how likely is the nature of this content only found in Mark given Matt, Luke, or some other gospel’s priority. I think a safe bet would also be 40% for Matt/Luke and an unknown for John since there is very little that John and Mark share, and unknown given some other unknown gospel. The content only found in Mark has no bearing on John or some unknown gospel since they don’t seem to share any sort of word-for-word matches. So John and any sort of catchall will still be agnosticism.

So our Bayes’ formula is like so:

P(Mark | COFIM) = P(COFIM | Mark) * P(Mark) / [P(COFIM | Mark) * P(Mark)] + [P(COFIM | Matt) * P(Matt)] + [P(COFIM | Luke) * P(Luke)] + [P(COFIM | John) * P(John)] + [P(COFIM | ~H) * P(~H)]

P(Mark | COFIM) = 60% * 26.08% / [60% * 26.08%] + [40% * 17.39%] + [40% * 17.39%] + [50% * 17.39%] + [50% * 26.31%]

P(Mark | COFIM) = 15.65% / [15.65%] + [6.95%] + [6.95%] + [8.69%] + [8.69%]

P(Mark | COFIM) = 15.65% / 46.95%

P(Mark | COFIM) = 33.33%

So given the content only found in Mark, P(Mark) is now 33.33%. P(Matt) and P(Luke) are 17.2%, and P(John) and P(~H) are 18.51%. I would bet that if we looked at the content only found in John or a heretical gospel as compared to Mark, these would increase in probability.

3. Primitive/Unusual Language in Mark; Editorial Alterations

Regarding verses where Mark differs from Matthew and/or Luke, it is often easier to see why Matthew or Luke would alter Mark than the reverse. For example, Matthew 20:20 eliminates a criticism of the disciples found in Mark 10:35 and later verses. Matthew 8:25 and Luke 8:24 both eliminate disrespect towards Jesus from the disciples in Mark 4:38.

Mark’s Jesus often seems more human than Matthew’s. Davies and Allison list a number of passages where Mark but not Matthew portrays Jesus as emotional (e.g. Mark 1:41, cf. Matthew 8:3), ignorant of some fact (e.g. Mark 6:37-38, cf. Matthew 14:16-17), or incapable of some action (e.g. Mark 6:5, cf. Matthew 13:58).

It is argued that it is easier to see why Matthew would edit Mark to make Jesus more divine and more powerful, than why Mark would edit Matthew to weaken Jesus.

[…]

Mark’s Greek is more primitive than the other Gospel writers. Often, Luke or Matthew will state a parallel Jesus quotation much more eloquently than Mark. In addition, Mark occasionally uses an unusual word or phrase where Matthew uses a common word. It is argued that this makes more sense if Matthew was revising Mark, rather than the reverse.

In addition, Mark is the only author who quotes Aramaic words and phrases which may have been the actual words of Jesus. He alone gives the words Boanerges (3:17), Ephphatha (7:34), Talitha cum (5:41), Abba (14:36) and the Aramaic form of Eli in the cry, Eloi, Eloi lama sabachthani (15:34). It has been argued by Geza Vermes that these quotations indicate a closeness to Jesus not shown in the other Gospels.

I would like to add that my own observation about why Mark would include the (Aramaic) redundancies Bartimaeus and Abba, owing to the idea that Mark invented the character Barabbas. So for example, Bartimaeus is an unusual word/name in that it is a mix of an Aramaic prefix (bar) and a Greek name (Timaeus). Mark then, for reasons unknown to at least Matt and Luke, explains that Bartimaeus means son of Timaeus (Mk 12.46). The Greek doesn’t actually have a parenthetical, it just says “the son of Timaeus Bartimaeus”. Matt and Luke leave out the redundancy, with Luke and John removing the name altogether (and John expanding the now anonymous blind man’s healing story to an almost entire chapter [John 9]).

Next in the sequence, Jesus has a redundant prayer in the garden of Gethsemane, where he says “Abba, father…” at Mk 14.36. This still follows Mark’s pattern of defining the non-Greek name in the immediate context it is written in. And again, Matt and Luke leave out the supposed redundancy. So by this point, the reader, if they are paying attention, has been told by Mark that “bar” means son of and “Abba” means father.

We are then introduced to a character named Barabba at Mk 15.7, seemingly to set up a human version of the Yom Kippur ritual. Two men who are the “son of the father”: One gets released, the other is slaughtered.

There are many instances of these, like the Wikipedia article says, where Mark has unusual words/phrases that Matt and Luke leave out. Almost every new “verse” in Mark’s Greek gospel begins with either εὐθύς::euthus (immediately/straightaway) or καί::kai ([strong] and). Imagine reading a book where every new sentence started with “immediately” or “and”. Quite normally a new editor would make the text more readable; Matt and Luke leave all of those immediatelys and ands. Furthermore, other gospels like John retain Mark’s “unusual” character or town names (e.g. Barabbas; Bethany). So, assuming that Mark wrote these unusual names first, this is strong evidence for Markan priority. Assuming some other gospel wrote the normal version first and Mark put the unusual names in a poetic/allegorical context seems highly unlikely.

So for another example, Bethany is Aramaic for house of misery. Sure, the name sounds nice in English, but why would someone name a town that in Aramaic? This is where, in Mark, Jesus is in the house of a man with leprosy and mourned over by a nameless woman (who Jesus ironically says will be remembered forever for her deeds). It is also where Jesus mentions how the poor will always exist. Luke, on the other hand, changes the house to that of a Pharisee in a town called Nain, with the woman now described at sinful.

It seems unlikely that Mark would insert these poetic/allegorical names and towns after they had been introduced by other gospel authors, whereas it seems highly likely that Mark would have created these weird names in his gospel for his narrative/artistic purpose and others wouldn’t know what they meant. So P(Unusual | Mark) = 80%, whereas for other gospels it would be very improbable, I’d say around 5%. John uses less of the unusually named people and places so his gets a bit less of a hit; maybe 20%.

Now, if there was some heretical/unknown gospel that existed before Mark that Mark himself was using that also included these weird/unusually named people/places, that uncertainty should also be included in this analysis… which is still agnosticism on my part.

So our Bayes’ formula is like so:

P(Mark | Unusual) = P(Unusual | Mark) * P(Mark) / [P(Unusual | Mark) * P(Mark)] + [P(Unusual | Matt) * P(Matt)] + [P(Unusual | Luke) * P(Luke)] + [P(Unusual | John) * P(John)] + [P(Unusual | ~H) * P(~H)]

P(Mark | Unusual) = 80% * 33.33% / [80% * 33.33%] + [5% * 14.81%] + [5% * 14.81%] + [20% * 18.51%] + [50% * 18.51%]

P(Mark | Unusual) = 26.66% / [26.66%] + [0.74%] + [0.74%] + [3.7%] + [9.25%]

P(Mark | Unusual) = 26.66% / 41.11%

P(Mark | Unusual) = 64.86%

So given the unusual people/place names, P(Mark) is now at 64.86%. P(Matt) and P(Luke) are 1.8%, P(John) is 9%, and P(~H) is 22.52%

4. Fatigue

[There are] a number of occasions where it appears that Matthew or Luke begin by altering Mark, but become fatigued and start to copy Mark directly, even when doing so is inconsistent with the changes they have already made. For example, Matthew is more precise than Mark in the titles he gives to rulers, and initially (Matthew 14:1) gives Herod Antipas the correct title of “tetrarch”, yet he lapses into calling him “king” at a later verse (Matthew 14:9), apparently because he was copying Mark 6:26 at that point.

Another example […] is Luke’s version of the feeding of the multitude. Luke apparently changed the setting of the story: whereas Mark placed it in a desert, Luke starts the story in “a town called Bethsaida” (Luke 9:10). Yet later on, Luke is in agreement with Mark, that the events are indeed in a desert (Luke 9:12). Luke is here following Mark, not realising that it contradicts the change he made earlier.

I think this is also pretty strong evidence in favor of Markan priority. It also sort of bleeds into the previous line of evidence about alterations. As a matter of fact, there’s one additional piece of editorial fatigue between Matt and Luke that makes Luke even less likely to have been written first than Matt. From Mark Goodacre’s Case Against Q:

The same phenomenon of editorial fatigue occurs also in double tradition material, where the evidence suggests that Luke is secondary to Matthew. In the Parable of the Talents / Pounds (Matt 25.14-30 // Luke 19.11-27), Luke, who loves the 10:1 ratio (Luke 15.8-10, Ten Coins, one lost; Luke 17.11-19, Ten Lepers, one thankful, etc.) begins with a typical change: ten servants, not three; and with one pound each (Luke 19.13). Yet as the story progresses, Luke appears to be drawn back to the plot of the Matthean parable, with three servants, “the first” (Luke 19.16), “the second” (Luke 19.18) and, remarkably, “the other” (Luke 19.20, ο ετερος). Moreover, the wording moves steadily closer to Matthew’s as the parable progresses, creating an internal contradiction when the master speaks of the first servant as “the one who has the ten pounds” (Luke 19.24), in parallel with Matthew 25.28. In Luke, he does not have ten pounds but eleven (Luke 19.16, contrast Matt. 25.20).

So this will be one instance where Matt and Luke shouldn’t have the same conditional probability. Based on this type of evidence, Luke is even further removed than Matt from Mark.

What is the probability that Mark would erase editorial fatigue in the direction of factually wrong information? Or in other words, what is the probability of having inconsistency given that Mark was written first? What is the probability that someone copying someone else’s work would suffer from editorial fatigue (inconsistency) by introducing inconsistent facts and inconsistent parables? Assuming Mark is written first, we would expect consistency. So P(Consistent | Mark) = very probable, or 95%. P(Consistent | Matt) is improbable, or 20%. Luke has more inconsistency than Matt based on Goodacre’s arguments (Luke’s inconsistency includes redaction inconsistency on both Mark and Matt’s work), so Luke is necessarily lower at 5%. On the other hand, other gospels like John or Marcion have no inconsistency (that we know of for the apocryphal/incomplete gospels) so I leave them at agnosticism at 50%.

Now the Bayes formula is:

P(Mark | Consistent) = P(Consistent | Mark) * P(Mark) / [P(Consistent | Mark) * P(Mark)] + [P(Consistent | Matt) * P(Matt)] + [P(Consistent | Luke) * P(Luke)] + [P(Consistent | John) * P(John)] + [P(Consistent | ~H) * P(~H)]

P(Mark | Consistent) = 95% * 64.86% / [95% * 64.86%] + [20% * 1.8%] + [5% * 1.8%] + [50% * 9%] + [50% * 22.52%]

P(Mark | Consistent) = 61.62% / [61.62%] + [0.36%] + [0.09%] + [4.5%] + [11.26%]

P(Mark | Consistent) = 61.62% / 77.83%

P(Mark | Consistent) = 79.1%

So given the consistent parables/pericopae, P(Mark) is now at 79.1%. P(Matt) is at 0.46%, P(Luke) is at 0.11%, P(John) is 5.78%, and P(~H) is 14.46%

Final Judgement

The probability of Mark being written first given all of the evidence I presented here: 79.1%, or “probable” range. The thing to remember is that this isn’t meant to give the illusion of mathematical precision but for simply making uncertainty less ambiguous; meaning that given all of the evidence presented here, I’m still actually just saying that it’s “probable” that Mark was written first, not that there is a 79.1% chance that Mark was written first. The less ambiguous our probabilities are (e.g. “highly unlikely”) the more we can actually rationally update our probability estimates based on evidence, e.g. “this x is highly unlikely, but given this evidence y makes it much more likely that x has happened” is too vague to update rationally. And it’s entirely possible that someone could disagree with my subjective conditional probabilities, but I think the main point would be that these evidences favor Markan priority; the crucial bridge is the Bayes Factor that is produced. So maybe I think that the unusual names strongly favor Markan priority, whereas a conservative scholar might think they only slightly favor Markan priority, e.g. they think that P(Unusual | Luke) is 40% and not 5%. Either way, it is an increase in evidence for Markan priority. The real point of disagreement would be if someone thought these evidences were all actually evidence against Markan priority. Meaning that they thought Bayes Factor in each of these cases was less than 1.

And even in that case, two rationalists can combine likelihood ratios to come to an agreed likelihood. But that’s a bit complicated 🙂 (though I might make a post about that in the future).

So this is a really quick scan of the evidence in favor of Markan priority. Ideally, someone would go over each singular piece of evidence (e.g. finding a likelihood ratio comparing competing hypotheses for primacy given the “who struck you?” verse at Matt. 26.67-8 // Luke 22.63-4 that’s missing at Mark 14.65) and do a Bayesian update on each data point instead of grouping them together like I did above. But I guess if someone did that, that would be very data intensive, be a few hundred pages, and probably count as some sort of thesis publishable in a peer-reviewed journal, not a blog post!

But yeah, this isn’t the end of the story. Next, I’ll go over some evidence in favor of Matthew’s priority using some Bayescraft.

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Posted by on October 21, 2013 in Bayes, bayes theorem and the synoptic problem

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