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 H_{DEAD BODY RAISED BY YAHWEH} with H_{DEAD 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 | H_{DEAD BODY RAISED BY YAHWEH}) for the strength of his argument. Unfortunately, this also works for P(E | H_{DEAD 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 H_{DEAD 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 | H

_{DEAD BODY RAISED BY YAHWEH}) * P(H_{DEAD BODY RAISED BY YAHWEH})

Denominator for Craig’s argument:

P(E | H

_{DEAD BODY RAISED BY YAHWEH}) * P(H_{DEAD BODY RAISED BY YAHWEH})

+ P(E | H_{DEAD BODY REPLACED BY ALIENS}) * P(H_{DEAD BODY REPLACED BY ALIENS})

+ P(E | H_{OTHER HYPOTHESES}) * P(H_{OTHER 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.