Home » Posts tagged 'Bayes’ theorem'

Tag Archives: Bayes’ theorem

Josephus, Jesus and Bayes’

This entry is part 10 of 11 in the series On the Historicity of Jesus

Continuing with my theme of bouncing all over the place, according to whim and fancy, next up is the Testimonium Flavianum.

In the late first century, historian Flavius Josephus published his book Antiquities of the Jews, which is a history of the Jewish people. In the book are found two references to Jesus Christ whose authenticity have been questioned by scholars. I’ll just briefly indicate why for the first of these references.

Antiquities Book 18 3.3

Josephus was a Jew, not a Christian, so you’ll notice that phrases such as “He was the Christ”, and “if it be lawful to call him a man”, cannot possibly be authentic. Since all of our extant manuscripts come from Christian sources it would seem that phrases such as these, at the least, have been added by Christians. Now, some scholars want to argue for an authentic core to the passage, but I find this to be nothing more than wishful thinking.

For a good summary of the arguments regarding authenticity, pro and con, I suggest this article on Peter Kirby’s Early Christian Writings. The arguments that I find conclusive are 6(missing from table of contents), 8 (doesn’t fit well in the surrounding material), 9 (not cited in works of early church fathers and should be if it was there), and 11 (sounds very much like the language of Eusebius, who may have been the forger himself) on the con side. For more on number 11 see this blog post by Ken Olson brought to my attention in the review of OHJ by Nicholas Covington.

Devilish Details
The rest of this post is just me playing around with some ways to visualize the task of  coming up with likelihood ratios. I’m not sure if this will be at all useful to anyone so you may want to skip it (no, trust me, you probably do). I’m just experimenting, really.  As I said above I consider the Testimonium completely spurious but find it to be a convenient starting point for the discussion below.

I’ve said before that what we are doing when developing our likelihood ratios is imagining the total space of possible worlds under the hypotheses we are comparing. We are then comparing how often our evidence shows up under each. We don’t mean the very specific details. For instance, we don’t really care that the Testimonium Flavianum shows up specifically in the 18th book of the Antiquities, that isn’t relevant. So we are abstracting away some of the details and want to know which possible worlds have evidence ‘like’ ours.

  • Historicity Possible Worlds
    • Contains passage ‘like’ Testimonium (x%)
    • Doesn’t contain passage ‘like’ Testimonium (100 – x%)
  • Mythicism Possible Worlds
    • Contains passage ‘like’ Testimonium (y%)
    • Doesn’t contain passage ‘like’ Testimonium (100-y%)

Our ratio then will be y divided by x. Obviously, it will matter quite a bit what our evidence is ‘like’. For example, if ‘definitely authentic’ is one of the characteristics the passage has, x and y will be very different than if the passage is inauthentic.

Our intuition might be that an interpolated passage (i.e. not authentic) doesn’t argue for either historicity or mythicism. If we take a step back though, and look at the situation before we decide on authenticity, we might think differently. I’ll show you what I mean.

  • I-HistoricityPWs
    • A- No passage in Josephus
    • B- Passage present but entirely interpolated
    • C- Passage present and has, at minimum, an authentic core
      • 1- Info gleaned from Christian report only
      • 2- Info independent of (or mostly independent of ) Christian reports
  • II-MythicismPWs
    • A- No passage in Josephus
    • B- Passage present but entirely interpolated
      • 1- Info gleaned from Christian reports only

I want to briefly defend dividing up the possible worlds as I have done in A, B and C.  As long as the items in any subtree add up to 100% any division should theoretically be as good as any other. Before we divided them up into ‘like our evidence’ and ‘not’, here I add a more finely grained view. Now, not every way of dividing up the space will be as useful, but there is nothing wrong with doing things this way. In fact, in this case, it aligns better with how we naturally want to approach the problem. We would like to figure out if the passage is an interpolation(or not)  and then figure out how often interpolations show up under each theory.

Getting back to the outline above we should first notice how difficult it would be to actually estimate the numbers we need. We are basically floating free here. (I think) we have no actual, empirically derived,  numbers to use. Still, perhaps we could say something about the relationships between them. It seems like I.C. should be bigger than II.C. for the simple reason that there are more ways for an authentic passage to show up under historicity. This means that I.A. or I.B., or both, have to be smaller than their counter parts in II. So which one is it?

Our initial intuition might have been that an interpolated passage was just as likely under either theory and so the extra occurrences of an authentic passage are all coming out of the “No passage” bucket. I would be more inclined to think it would diminish a little of each. There is less need to interpolate under historicity and therefore it would occur less. I think whatever the ratio of II.A. to II.B. is should be maintained under I. Redrawing the tree in a slightly different way, and adding some actual numbers (pulled straight out of nowhere), might help visualize what I’m saying

  • I-HistoricityPWs
    • A – Passage present and has , at minimum, an authentic core (4%)
      • 1 – Info gleaned from Christian report only (50%)
      • 2 – Info independent of (or mostly independent of ) Christian reports (50%)
    • B – No authentic passage (96%)
      • 1 – No passage at all (75%)
      • 2 – Entirely interpolated passage (25%)
  • II-MythicismPWs
    • A – Passage present and has , at minimum, an authentic core (2%)
      • 1 – Info gleaned from Christian report only (100%)
    • B – No authentic passage (98%)
      • 1 – No passage at all (75%)
      • 2 – Entirely interpolated passage (25%)

So B. takes up a slightly different amount of space in the over all possibility space under each hypothesis, but is split up between ‘no passage’ and ‘interpolation’ the same.

This all means that an interpolated passage would actually end up favoring mythicism. Not so much because the passage is interpolated, but because it falls under the no authentic passage scenarios which happens more often under mythicism. This fits in with what I said in a previous post where I felt that each failure to find a mention of Jesus should at least count somewhat against historicity.

Does this seem right, though? The arguments I gave seem reasonable but it definitely gives me a healthy respect for the complexity of the situation. If I’ve got the relationships right, the numbers themselves are probably less than justified (I just made them up). My thinking was that there was probably a low chance for inclusion by Josephus even if he has heard of Christians but that, given the status of Josephus, there would be a strong motive to fill in any silences found in his work.

What’s It Like?

More important is the case where we can’t narrow things down to one branch but favor one over the other. In the case of the Testimonium I’m convinced we can proceed as if our passage is interpolated; I don’t think there is much chance it has even an authentic core. For the other passage in Josephus however I’m not as confident of that conclusion.

Just adding together both the interpolated and authentic options won’t do in this less clear cut scenario. Some of those possible worlds will have unmistakable evidence one way or another, and they aren’t ‘like’ a passage that could be either. Besides we may heavily favor one, interpolation say, but still give authenticity a reasonable chance of being true.

Jeffrey Conditionalization is one way forward. Basically we just do a weighted average, which requires us to estimate how likely each branch is. The way it would work in this case is we would calculate our posterior probability as if we know the passage is authentic, and then multiply by the percent chance we think that is the case. Then add to that the calculation, done as if we know it isn’t authentic, multiplied by the percent chance we think that is the case. Too many words… an example:

Let’s say the posterior probability of mythicism is 0.333 , if the passage is authentic, and we started  with a flat prior (i.e. P(h) = 0.5 = P(~h) ). But lets say we only think there is a 5% chance the passage is authentic so we multiply: 0.333 *  0.05 = 0.017.

Now let’s say the posterior probability for an interpolation (we know there isn’t “no passage”) is 0.505. We think it is 95% likely this is the case so we multiply: 0.505 * 0.95 = 0.48

Now adding those two numbers together we come up just shy of 0.5. Surprised? Those are the actual number you will get, by the way, if you plug the numbers in my outline into your favorite Bayes calculator. What is happening is that an authentic passage is a much better indicator than an interpolated one, so even though I gave authenticity a mere 5% chance of being true, it was enough to bring the posterior below 50%.

The final number is sensitive to how likely we think it is for an authentic passage to show up. For example, if I change II.A. to 5% and I.A. to 10%, and again assume a flat prior, we get a final posterior slightly above 50%. If we go to 10% and 20% then the posterior will be about 52%. All in all calling it a wash seems fair since I have no idea what those actual numbers should be, however do note that if I give authenticity as little as a 20% chance of being true we are back under 50% even with the larger estimates for I.A and II.A.

The lesson here is that a characteristic that sharply distinguishes between two theories can sway our results even if we think it has a low probability of being true.


Moving on, the strangely worded phrase in the heading is the other reference found in Josephus, this time in Antiquities 20.9.1. I again recommend Peter Kirby’s site for the arguments for and against this passage.

The phrase does look a bit odd, with Jesus coming first and James last, but Peter Kirby thinks it is just another grammatically correct way to identify James. I also find items 4 and 5 in favor of authenticity important.  Against authenticity I find items 4 through 6 sway me to consider interpolation more likely than authenticity, but not by much.

Richard Carrier has published a peer-reviewed article on this passage where he suggests the passage was an accidental interpolation. That kind of thing happened a lot back then. They had no method to distinguish between a marginal note and a phrase that had been left out during copying of the manuscript, so often a marginal note would be included in the text the next time it got copied. To read the article you can buy Carrier’s book Hitler Homer Bible Christ or read a summary of the argument here.

He offers a very plausible scenario for how an interpolation might have occurred. Still I don’t find the arguments quite as convincing as those against the Testimonium. I’m going to proceed by assuming there is a 25% chance of the passage being authentic.

  • HistoricityPWs
    • No passage (90%)
    • Interpolated passage (5%)
    • Authentic passage (5%)
  • MythicismPWs
    • No passage (94.999%)
    • Interpolated passage (5%)
    • Authentic passage (.001%)

With the Testimonium I see interpolation as a purposeful action designed to fill a void. Here any interpolation is going to be accidental. Therefore I’ve lowered the rate of occurrence and kept it at a fixed rate. I could have structured things the same as before but this way gives me a nice even fraction which I would probably round off to anyway (I checked).

Using these numbers and the technique I used in the previous section:

Assuming an authentic passage, and flat priors gives us essentially zero. And, zero multiplied by 25% is still zero.

Assuming an interpolated passage, and flat priors gives us 0.5. Multiplying by 75% gives us: 0.375

So if this were our only piece of evidence we would have 37.5% chance for myth and 62.5% chance for historicity. Since I’ve been using ratios instead of percentages, dividing 37.5/62.5 gives 0.6… or a 3 to 5 ratio in favor of historicity. The Testimonium I found to favor neither theory (i.e. the ratio was 1/1) so 3/5 is our final answer.


Josephus: 3/5

OHJ – Tally Sheet

A Mathematics of History?

This entry is part 2 of 11 in the series On the Historicity of Jesus

Before diving into the contents of Richard Carrier’s On the Historicity of Jesus (OHJ) I need to say a bit about the method he uses to organize his conclusions. Prior to this book Carrier wrote a book called Proving History. It was entirely concerned with what methodology he would be using to answer the question of Jesus’s historicity. It included a primer on historical inquiry in general, a critique of the methods biblical scholars use to “extract” historical information about Jesus from the gospels, and a defense of using Bayes’ Theorem in historical inquiry.

The Critique
Biblical scholars for some time have used various criteria which they claim allows them to separate the fictional elements of the gospels, added by the church, from the historical elements. Or, at least they claim it increases the likelihood that some element is historical. For example the criterion of embarrassment says if an element of the story would be embarrassing to the church it is not likely they would have invented it.

The criteria have come under strong criticism within the field. The logic of them often is not sound and scholars apply them in situations where they aren’t applicable, or apply them inconsistently. Indeed the history of their use is not very encouraging as each scholar who sets out to paint a picture of “the true historical Jesus” comes back with a different picture. Carrier cites a lot of this critical scholarship and adds his own critique. I’ll just give one example and summarize the analysis given by Carrier (which I agree with).

One of the supposed bedrock truths scholars have extracted from the gospels is that Jesus was baptized by John the Baptist. The criterion of embarrassment is invoked. Here is Jesus being portrayed as in some sense subordinate to John. In addition John’s baptism was for the remission of sin and wasn’t Jesus supposed to be without sin? Thus, the reasoning goes, this must have been something everyone knew about and it just couldn’t be avoided. Mark was written first and Matthew and Luke copied much of their material from him (often word for word, or nearly so). Matthew and Luke have tinkered with this story to make it less embarrassing.

It is a reasonable argument but there are several problems. The later gospel authors seem to have no compunction about altering stories to fit their agendas. They omit elements when it suits them. Why would Mark be any different? If he was embarrassed by this story why did he include it in the first place? In fact Mark shows no signs of embarrassment at all in telling this story. Let’s not forget that Mark was written roughly 35 years after Jesus would have lived. So are we to imagine the story of Jesus being baptized by John circulated for three and a half decades and no one noticed it was embarrassing? No one had already tinkered with the story before it reached Mark (and thus he would have already been reporting the apologetic in his gospel)? Suddenly, after Mark wrote his gospel, everyone finally noticed it was embarrassing and scrambled to explain it!

When you think it through it is a rather silly scenario. In fact, the actual sequence of events suggests quite the opposite of what the biblical scholars have concluded. Mark being completely unembarrassed by the story followed by signs of embarrassment after Mark wrote suggests that no one had heard the story before because Mark invented it! Either that or it simply wasn’t embarrassing to those that came before, but if that is case the criterion of embarrassment doesn’t apply!

I’ve gone one far longer than I intended so I’ll just leave it at that. I’ll just say I think Carrier, and the critics from within the field, have successfully called into question the validity of criteria based methods.

Bayes’ Theorem
The challenge facing anyone examining a complex historical scenario is this. Each individual artifact or document could have had several causes and in most cases you cannot rule out all but one of them. So how do you evaluate your overall thesis if each piece of evidence only partially supports it? Also, it will often be the case that some pieces of evidence are more likely on one theory but other pieces suggest another (while not ruling out the first one). Clearly what is needed is some way to state how much weight each piece of evidence carries, and by how much it favors one theory over another. Finally, there needs to be some way to combine those individual judgments in a fair and rigorous fashion.

That’s where Bayes’ Theorem comes in. Bayes’ Theorem is an equation in probability theory specifically geared towards comparing one hypothesis with another (or, rather, one hypothesis with its negation). If we can state our evaluations of the evidence in terms of probabilities (or ratios of probabilities) then we can use Bayes’ Theorem to combine them, giving us a number at the end that states how likely it is the theory we are considering is the cause of the evidence we have. Obviously that number is entirely dependent on how sound our evaluations were but at least it gives us a principled way to combine them while at the same time allowing others to see how we’ve weighted the evidence.

Carrier defends the use of Bayes’ Theorem in Proving History and responds to most of the common objections people advance. I’m not going to spend a lot of time on that material. I think it is an interesting subject and I may discuss it during the course of this series but for now I think it is enough to say that using Bayes’ Theorem seems reasonable enough. In OHJ Carrier simply tallies how strongly he thinks a piece of evidence favors one theory or the other, e.g. 5/4 in favor of mythicism, or 1/1 if it doesn’t favor either, etc. To combine them one simply multiplies the individual factors together to get a final ratio that tells you which theory is indicated by the evidence and how strongly.

So what do those ratios represent? I like to think of it in terms of the scientific method we all learned in school. The idealized scientific method is often presented as hypothesize, enumerate predictions, gather data, reach conclusions. The ratios represent the ‘enumerate predictions’ part of this picture. What we are doing is taking a hypothesis as a given, i.e. assuming it is true, and then asking ‘of all the possible worlds that could result with this hypothesis being true, what percentage of them will contain the piece of evidence we are evaluating?’ Then we do the same for the hypothesis we are comparing it to (or the combination of all hypotheses that aren’t the first one). Dividing the first by the second gets us our ratio. So if our ratio is 2/1 in favor of historicity, for example, we are saying a historical Jesus will produce the evidence twice as often as a mythical one will.

It should be noted that there is a certain level of generality involved here. If examining a document for instance, we aren’t asking how often that specific document with exactly the words it contains will arise. We are asking instead about general features of the document. Carrier mounts a defense of this approach in Proving History and I think it is generally sound, however I would point out it introduces another place where subjectivity can enter the picture. Which features are important and which features can be abstracted away?

There is one more feature of Bayes’ Theorem that must be mentioned. When you use it you must estimate the prior probability of the hypothesis you are testing. That is, before you even begin looking at evidence how likely is your hypothesis, based on your more general knowledge of the world? Carrier uses a scale, developed by Rank and Raglan, of traits often found in hero tales, on which Jesus scores very high. Though historical people sometimes have several of the traits, that arise in legends about them, no historical individuals score nearly as high as Jesus and the high scoring mythical beings. From this Carrier justifies starting with a prior probability that Jesus was historical of 33% (i.e. 1 in 3, or 2/1 against).

I’m not going to examine the issue of prior probability very closely at the moment. Nor will I be relying on his 33% estimate. The more important question is which hypothesis does the evidence favor, and by how much? If the evidence favors one hypothesis much more strongly than the other, then it won’t really matter much what prior you started with; it will be overwhelmed by the evidence. If the evidence is fairly close, then determination of the prior will become more important and can be pursued then.