Bayesian Analysis of Craig Keener, “Otho: A Targeted Comparison”

[Recently I wrote a critical review of Christian scholar Craig Keener’s new volume Biographies and Jesus: What Does It Mean for the Gospels to Be Biographies?, with emphasis on chapter 6–“Otho: A Targeted Comparison of Suetonius’ Biography and Tacitus’ History, with Implications for the Gospels’ Historical Reliability”–which is written by Keener himself. Ancient historian Richard Carrier sent me some further analysis, which makes both a deductive and inductive critique of Keener’s arguments. Carrier’s feedback can be found below. -MWF]

Applying Bayes’ Theorem to your article’s point:

Keener says we can be sure Suetonius et al. worked from sources, because they say they worked from sources. Then he says we can assume the same of the Gospels, because the Gospels have other similarities to Suetonius et al., except for that one.

This is a straightforward fallacy of false generalization. “All X’s did Y, and all Y’s entail doing Z, therefore all X’s did Z” does not lead by any valid logical inference to “The Gospels are an X,” precisely because the Gospels did not do Y (so the first premise in the argument fails to obtain). So any other similarities there may be are analogically irrelevant to whether the Gospels did Z. Only doing Y can entail Z. He would need to find examples of texts that we can be certain did Z, without doing Y (Y being “naming and discussing sources”). Without arguing in a circle.

So for the deductive logic.

But an apologist will insist it’s inductive. But then Bayes’ Theorem enters.

Keener’s argument that “we can be sure Suetonius et al. worked from sources” has this form:

The probability that they would say Y and not have done Z is low; therefore, given Y, the probability they did Z is high. But Keener has no evidence this relation holds for anything other than Y.

Where Y is in e, as are all other similarities between Suetonius et al. and the Gospels, then:

P(Z|e) = P(Z)P(e|Z) / [ P(Z)P(e|Z) + P(~Z)P(e|~Z) ]

Suppose we break the evidence into just Y, and then X for all the other parallels.

For just Y:

P(Z|Y) = P(Z)P(Y|Z) / [ P(Z)P(Y|Z) + P(~Z)P(Y|~Z) ]

Assume we’re neutral on the prior (we might not be, but that depends on other arguments and we are just analyzing this one), so that P(Z) = P(~Z), then:

P(Z|Y) = P(Y|Z) / [ P(Y|Z) + P(Y|~Z) ]

Which –> 1 as P(Y|~Z) –> 0.

That’s Keener’s argument.

But when he turns to the Gospels, he falsely treats X as if it were Y. But that doesn’t work. His argument from X is:

P(Z|X) = P(Z)P(X|Z) / [ P(Z)P(X|Z) + P(~Z)P(X|~Z) ]

And with a neutral prior that’s:

P(Z|X) = P(X|Z) / [ P(X|Z) + P(X|~Z) ]

Keener presents no evidence that that –> 1 as P(X|~Z) –> 0, nor any evidence that P(X|Z) is even high.

What instead he does is argue:

P(Z|Y&X) = P(Y&X|Z) / [ P(Y&X|Z) + P(Y&X|~Z) ]

Which gets him the Y result, then he uses the same argument for the Gospels, but “forgets” the Gospels don’t have Y. He is thus conflating Y with X. To argue X correlates with Z in the absence of Y requires actual evidence that that is ever the case. He presents none. Presenting examples that correlate Z with Y&X simply does not constitute evidence that Z correlates with Y. That’s the generic Bayesian analysis of the fallacy of false analogy in a nutshell.

Conversely, you point out that the generic similarities in X are actually known or credible properties even of fiction, so that in fact the evidence there is actually argues *against* any distinct correlation between X and Z (it may be at best 0.5, such that P(Y&X|Z) = P(Y&X|~Z)), so it’s even worse than Keener having no evidence that X correlates with Z; the evidence actually is against there being any such correlation, at least in any reliable sense. That’s the “at best” 0.5 correlation; but it’s possibly worse, if the absence of Y is telltale of fiction, another argument of yours. And that need not be a correlation of 1, it could be, say, 0.8, allowing 20% of examples of no-Y still being Z texts. Look what happens when you are even that generous (and still using a neutral prior as if no other considerations mattered, which we know isn’t the case), assuming no correlation exists between no-Z texts containing X:

P(Z|X) = P(X|Z) / [ P(X|Z) + P(X|~Z) ] = 0.2 / (0.2) + (0.5) = 0.29.

If no-Z texts typically contained X, it’s even worse. Only if no-Z texts rarely contain X would it get better; but that would require the very evidence Keener doesn’t present: that X correlates with Z.

Introducing Y also changes the result, of course. But that’s precisely what the Gospels don’t do. Likewise any other generic factor that might up the odds of Z; which would need to be demonstrated as doing so in other texts (and without circular argument).

-Richard Carrier

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Understanding the Spirit vs. the Letter of Probability

My apologies for there being so little activity on Civitas Humana for a long while now. I have been insanely busy and stressed lately (both professionally and personally), and I have only recently been able to catch up with blogging.

I’ll begin pumping some fresh blood into Civ by discussing Bayes’ theorem, the resurrection of Jesus, and why I think that, even with a non-zero prior (which is still very, very low) for the resurrection event, Paul’s letters and the Gospels are far too weak of consequent evidence to offset more probable (naturalistic or mundane) explanations for the same data.

Over a decade ago (March 2006) secular New Testament scholar Bart D. Ehrman debated Christian philosopher and theologian William Lane Craig about the evidence for the resurrection of Jesus (the transcript of the debate can be read here). Overall, the debate left me with the impression that Ehrman made a better case for naturalistic or non-paranormal explanations being more probable than a veridical resurrection event, with regards to the origin of the resurrection belief among Jesus’ disciples and the first generation of Christians. But there was one area where I think Craig scored a technical, though relatively minor point against Ehrman (as will be discussed below), and this was with regards to how Ehrman was defining a miracle event and conflating prior probability with posterior probability.

Lowder’s Summary of Craig’s First Rebuttal

Jeff Lowder on the Secular Outpost has made a useful summary of Craig’s critique of how Ehrman was defining a miracle event, and the arguments Ehrman had presented (in earlier publications) for why historians cannot argue that a miracle is the most probable explanation for a past event. This summary can be read in Lowder’s post “William Lane Craig’s Critique of Bart Ehrman on the Probability of Miracles.” Since the ensuing discussion involves Bayesian reasoning, you can read my essay “History, Probability, and Miracles” for a basic overview of how Bayesian logic works, if you are unfamiliar with the theorem.

Lowder begins by listing two published statements by Ehrman, which were quoted by Craig during the debate (bolding is my own):

(1) “Because historians can only establish what probably happened, and a miracle of this nature is highly improbable, the historian cannot say it probably occurred.”
(The Historical Jesus, part 2, page 50)
(2) “Since historians can establish only what probably happened in the past, they cannot show that miracles happened, since this would involve a contradiction — that the most improbable event is the most probable.”
(The New Testament: A Historical Introduction, page 229)
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In his response to these statements of Ehrman, Craig critiques his line of reasoning by arguing that Ehrman is conflating prior probability with posterior probability. The odds that a given individual may resurrect from the dead could, indeed, be very, very low. But if there is very, very good evidence that such a resurrection event has occurred, it may offset the low prior, and even outweigh alternative explanations, to degree such that Pr (R/B & E) > 0.5 (perhaps even by a wide margin, e.g., +0.9).
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Here is what Craig states in his own words (bolding is my own):
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“In other words, in calculating the probability of Jesus’ resurrection, the only factor he [Ehrman] considers is the intrinsic probability of the resurrection alone [Pr(R/B)]. He just ignores all of the other factors. And that’s just mathematically fallacious. The probability of the resurrection could still be very high even though the Pr(R/B) alone is terribly low.”
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Lowder likewise offers his own interpretation of Ehrman’s two quotations, and here is what he states regarding the first:
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“I am inclined to interpret (1) as the following claim: (1′) Pr(R/B) is so low that it is impossible, even in theory, for there to be sufficient evidence to confer a high final epistemic probability on R, i.e., Pr(R/B & E) > 0.5.”
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I would argue that to describe a miracle as an event that cannot be probable, even in theory, one would need to assign its prior probability a value of zero. And this is the same conclusion that Lowder reaches, when he states (bolding is my own):

The only way to reconcile (1′) with BT would be to assign Pr(R/B) a value of zero. If Pr(R/B) = 0, then it follows from BT that Pr(R/B&E;)=0. So, on the basis of (1) alone, as Craig has quoted Ehrman, I think it is premature to assume that Ehrman ‘just ignores all of the other factors.’ Maybe he does do that, but the quotation provided in (1) doesn’t show that. What I can say is that either Ehrman ignores all of the other factors or Ehrman assumes that historians must assign Pr(R/B) a value of zero. If the latter, then I think that is false.”

With regards to Ehrman’s second quotation, Lowder briefly states:

“Turning to (2), I don’t have much to say, other than I think Craig is 100% correct when he says that Ehrman ‘Confuses Pr (R/B & E) with Pr (R/B).’”

My Thoughts on Ehrman’s Quotes

At the end of this essay, I will make some suggestions for improving Ehrman’s arguments (which in spirit I think are correct, even if they may be formally invalid at parts). That said, I agree with Lowder’s conclusions on both accounts, for at least three reasons:

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