## William Lane Craig and the Base Rate / Prosecutor’s Fallacy

21 Feb

John Loftus posted a video of Dr. Freed’s critique of Craig’s use of Bayes’ Theorem in a debate with Bart Ehrman. Since this is related to my previous post on Bayes’ and the virgin birth of Jesus, I thought I would attempt a much simpler explanation for Craig’s error.

Craig is arguing that, given the historical reliability of the gospels (i.e. the “four facts” that he relies on for Jesus’ resurrection), we have a high probability of Jesus’ resurrection from the dead. Unfortunately, this does not follow simply due to the low base rate for resurrections from the dead. That is the basis of the base rate fallacy. What Craig really seems to be arguing is that the evidence we have, the four facts, make sense given that Jesus was raised from the dead. That may be true, but only relying on that is the Prosecutor’s Fallacy.

Here’s why, using Bayes’:

H = Jesus is raised!

E = four facts

P(H) = probability of Jesus being raised from the dead before looking at E.

P(E | H) = probability of the four facts given that Jesus is raised

P(E | ~H) = probability of the four facts given some other hypothesis

The numerator of Bayes’ is this: P(E | H) * P(H). What happens if P(H) is low? What happens to the numerator as P(H) “approaches” zero? That’s right: the numerator in total tends towards zero. Again, P(H) is the probability of being raised from the dead before looking at E.

Of course, a compounding problem is that ~H isn’t just “Jesus was not raised”. It’s any other hypothesis that makes sense of the evidence. What if instead of comparing H with ~H, we compared HDEAD BODY RAISED BY YAHWEH with HDEAD BODY REPLACED BY ALIENS? Again, the prior probability of alien body snatching seems to be equivalent with the prior probability of being raised from the dead. Both are extremely low. Alien body snatching would also make sense of the NT historians’ four facts; aliens could have projected images of Jesus to the disciples. This means that alien abduction theory is just as plausible as the resurrected by Yahweh theory. That is, if we ignore the base rate for both.

Like I said, Craig seems to be relying on the conditional probability P(E | HDEAD BODY RAISED BY YAHWEH) for the strength of his argument. Unfortunately, this also works for P(E | HDEAD BODY REPLACED BY ALIENS) or any other H that makes P(E | H) a high probability. Relying only on the conditional probability is a base rate fallacy. Furthermore, since these two conditional probabilities are equivalent, and HDEAD BODY REPLACED BY ALIENS is included in ~H and P(E | ~H), this means that the denominator of Bayes’, in this argument, will always be higher than the numerator, contrary to Craig’s assertion. Look at it this way:

Numerator for Craig’s argument:

P(E | HDEAD BODY RAISED BY YAHWEH) * P(HDEAD BODY RAISED BY YAHWEH)

Denominator for Craig’s argument:

P(E | HDEAD BODY RAISED BY YAHWEH) * P(HDEAD BODY RAISED BY YAHWEH)
+ P(E | HDEAD BODY REPLACED BY ALIENS)  * P(HDEAD BODY REPLACED BY ALIENS)
+ P(E | HOTHER HYPOTHESES) * P(HOTHER HYPOTHESES).

So let’s say the numerator is X%. The denominator would be X% + Y% + (100% – X% + Y%), which will always be larger than X% by itself. Craig’s logic would only work if H were a binary hypothesis.

Again, we would need more corroborating evidence besides Craig’s four facts to push the incredibly low prior probability of Jesus being raised from the dead to a level where it is rational to believe. So not only is this a base rate fallacy, it’s also an improper use of ~H. Remember, ~H is exhaustive and not just a dichotomy between itself and H. ~H could include raised by Horus, raised by Krishna, raised by aliens, raised by Zeus, raised himself (a la Marcion and the Marcionite God), swoon theory, NT historians are mistaken about the four facts, it’s all just a story invented by Mark, or any other hypothesis that makes sense of E.

Posted by on February 21, 2012 in Bayes

### 2 responses to “William Lane Craig and the Base Rate / Prosecutor’s Fallacy”

1. February 22, 2012 at 5:57 am

Thanks for posting this. You don't normally get a ton of comments, so I just wanted to say thanks for taking the time to write.

2. February 23, 2012 at 4:54 pm

Thanks! I normally post and comment on my phone so it takes a while to both post and keep track of comments

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