I would like to summarize the use Bart Ehrman made of a Bayesian argument, or tried to.
The occasion was a debate with William Lane Craig on the Resurrection's historicity:
http://www.reasonablefaith.org/is-t...or-the-resurrection-of-jesus-the-craig-ehrman
Ehrman's view was that a miraculous cause is always the
least likely explanation of an observed event. In fact, Ehrman claims that he can, in general, construct an explanation of an event, thoroughly
ad hoc, but because it is composed entirely of natural elements, it is more likely than a miraculous explanation, regardless of how far-fetched the construction is.
Note that this is a qualitative claim about probabilities. Nobody's putting a number on anything. Craig's rejoinder is that Ehrman has confused prior and posterior probability: the miraculous may be the least likely before a "miracle" is observed, but it can become the most likely afterwards. Ehrman doesn't effectively rebut.
Can Bayes'
Theorem, applied with the Bayesian
interpretation of it, help us to understand the controversy, and perhaps to resolve it?
Let us take Ehrman to have conceded that the observables of the resurrection have been observed (an empty tomb mostly). Let us also take Craig to have conceded that Ehrman's construction (a tomb robbery gone bad, basically) is more likely than a miraculous explanation
a priori. Finally, let us take both to have conceded that the other's preferred explanation,
if it were in fact true, would equally well explain the observed. (Craig and Ehrman differ on some details of that, but we are examining Craig's claim that Ehrman erred in his algebra, so we set those aside here and for the time being.)
Note that we have once again made only qualitative statements. No numbers appear here.
Among its many forms, Bayes'
Theorem can be written:
for uncertain hypothesis
h and observed event
e
p( h | e ) = p( h ) * p( e | h ) / p( e )
In the Bayesian
interpretation:
p( h | e ) is a measure of confidence in the truth of h given that e has been observed
p( h ) is the
a priori confidence in the truth of the hypothesis h
p( e | h ) is a measure of how well h would explain e if h were in fact true
p( e ) is the
a priori plausibility of seeing what was seen
Let us compare two hypotheses: r (Craig's miraculous Resurrection) and c (Ehrman's
ad hoc construction)
Note that p( e ) doesn't depend on h, and so it is the same for c as for r.
From what each party conceded, e has been observed, and
p( c ) > p( r ) ... Ehrman's construction is more plausible
a priori than Resurrection
p( e | r ) = p( e | c ) ... Either would explain equally well if it happned to be true
So:
let us define k = p( e | r ) / p( e ) = p( e | c ) / p( e ) >> 0 (it's probably a huge number, but all we need is that it is a positive number)
then, substituting into the Bayes' Theorem equations,
p( r | e ) = p( r ) * k
p( c | e ) = p( c ) * k
so, if p( c ) > p( r ), then p( c ) * k > p( r ) * k, or simply p( c | e ) > p( r | e )
Which is what Ehrman claimed, that the
ad hoc explanation of something we have observed is more credible than the miraculous explanation of it.
QED.
Now, this makes me a history-denying Bayesian activist, how?
