If you move in the same circles that I do then you’ve probably heard the following phrase many times, “Absence of evidence is not evidence of absence.” In and of itself this is totally true. In fact, it is just a special case of a well-known logical fallacy called an argument from ignorance.

One of the really cool things about using Bayesian methods when analyzing historical events (actually you could adapt the following to the example of the scientific method as well) is that you can quantify how improbable a certain absence of evidence is to make a sound argument. This allows you to conclude that a historical event actually did not take place based on the absence of evidence.

Now I could try to do this in some extremely abstract fashion, but it is so much clear to just show you in an example. There’s some good news and some bad news. The good news is that this post is being made near Easter, so the example is timely. The bad news is that some people might find the example highly offensive, because we will show using Bayesion inference that a certain event from the Gospels was entirely made up (or at least we can say with better certainty than we could ever hope for in our wildest imagination that this is the case).

This example, including the numbers, is entirely lifted from Richard Carrier’s book *Proving History*. This is not intended as plagiarism, but as I am not an expert in history I feel like randomly making up probabilities about how likely certain historical events are would just not make as convincing an example. Here’s the example: In the Synoptic Gospels (Matthew, Mark, and Luke) it is said that up to the death of Jesus the entire Earth was covered in darkness for three hours.

We want to figure out the probability that this was a historical event using the fact that there are no extra-Biblical accounts of this event happening. One thing to note is that there were civilizations all across the Earth in the first century who were already keeping copious records of bizarre astronomical phenomena that have survived to this time.

One very important thing to keep in mind when doing this example is the following. One might be tempted to make an argument about history vs supernatural events and so on. But the cool thing about this is that we don’t need to make any assumption about the occurrence of supernatural events to do this analysis. In fact, we could assume that supernatural events happen *all the time* and we will still come to the conclusion that this story was fabricated.

Let A be the statement that the Earth was covered in darkness for three hours. Let B be the event that we have no extra-Biblical accounts of this fact (I use the term “extra-Biblical” loosely to mean no sources that don’t admit they are referencing the Bible). We want to calculate P(A|B) the probability that the Earth was actually covered in darkness for three hours given the fact that we have no evidence for it.

The quantities that come up in Bayes’ theorem are the following: P(B|A), the probability we have no evidence of the event occurring supposing that it actually did occur. If we are exceedingly generous we can assign this probability at 1%. Note how high this percentage is though. Given our knowledge of surviving records of the time it is so mind-bogglingly unlikely that every civilization on the planet just accidentally missed something that would have scared them all out of their minds.

We also have P(A). This is slightly subtle, because in this case it represents not merely the probability that the event occurred, but is really the probability that the author of Mark (or one of the other Gospels which were probably just copying this detail) is telling the truth about the event occurring. More precisely, considering all the times that we know of (our “prior knowledge” as it was called in the previous post) of people telling us that the sun was blotted out how frequently did it actually happen (or less confusingly, when doing P(-A) how frequently did it turn out the story was made up). Being exceedingly generous again we’ll call this 1%.

Note we are not dismissing this on grounds of being a supernatural event (we’ve assumed for the purposes of this calculation that they happen all the time). The low number of 1% comes from the fact that we know of tons of examples in history where people tell stories like this one, but where we later find out they were made up. Lastly, we need P(B|-A) which is the probability of finding no external evidence for the event assuming the event was made up. This is so close to 100% that we may as well assign it a probability of 1.

Plugging everything in tells us that with at least (remember we were quite generous with the numbers) 99.99% certainty (re: there is a 99.99% chance that) the event never happened in history and was just made up by the authors of the Synoptic Gospels. And that is how Bayesian inference can lead to a sound argument from absence of evidence.

Of course, this should be an entirely non-controversial example because outside of a tiny few fundamentalist “scholars” who are clearly pushing an agenda, the fact that this even never happened in history has essentially unanimous consensus among all historians and Biblical scholars. So our result shouldn’t actually be surprising.

Being a huge fan of Bayesian reasoning myself, I very much liked your series so far. It reminded me of David Miller’s work; see

http://www2.warwick.ac.uk/fac/soc/philosophy/people/associates/miller/

In particular the slides “how probability generalizes logic”.

I was always hoping to find some time to blog about that, but maybe you’re now in a better position to do that.

The series isn’t over yet, right?

I want to get into Bayesian philosophy of math (and Bayesian proof theory), but before I do that I want to re-read what David Corfield wrote about it in

Towards a Philosophy of Real Mathematics. I remember it being great. Right now I’m at a conference, so it might be a late next week by the time I get around to it.