The current discussion among Jesus-deniers and mythicists over whether probability in the form of Bayes’s Rule can be used in historical research is more than a little amusing.

The current fad is largely the work of atheist blogger and debater Richard Carrier who despite having a PhD in ancient history likes to tout himself as a kind of natural science *cum* mathematics *cum* whachagot expert.

Carrier’s ingenuity is on full display in a recent book published by Prometheus (Buffalo, NY) in which he makes the claim that Bayes Theorem–a formula sometimes used by statisticians when dealing with conditional probabilities– can be used to establish probability for events in the past. That would make it useful for answering questions about whether *x* happened or did not happen, and for Carrier’s fans, the biggest* x* they would like to see answered (he claims ) is *Did Jesus exist or not? *

The formula looks something like this:

Let A_{1}, A_{2}, … , A_{n} be a set of mutually exclusive events that together form the sample space S. Let B be any event from the same sample space, such that P(B) > 0. Then,

P( A_{k} | B ) = |
P( A_{k} ∩ B )
P( A |

Invoking the fact that P( A_{k} ∩ B ) = P( A_{k} )P( B | A_{k} ), Baye’s theorem can also be expressed as

P( A_{k} | B ) = |
P( A_{k} ) P( B | A_{k} )
P( A |

Clear? Of course not. At least not for everybody. But that isn’t the issue because the less clear it is the more claims can be made for its utility. Its called the *Wow!* Effect and is designed to cow you into comatose submission before its (actually pretty simple) formulation, using the standard symbols used in formal logic and mathematics.

What is known by people who use Bayes’s theorem to advantage is that there are only certain conditions when it is appropriate to use it. Even those conditions can sound a bit onerous: In general, its use is warranted when a problem warrants its use, e.g. when

–

- The sample is partitioned into a set of mutually exclusive events { A
_{1}, A_{2}, . . . , A_{n}}. - Within the sample space, there exists an event B, for which P(B) > 0.
- The analytical goal is to compute a conditional probability of the form: P ( A
_{k}| B ). - You know at least one of the two sets of probabilities described below.
- P( A
_{k}∩ B ) for each A_{k} - P( A
_{k}) and P( B | A_{k}) for each A_{k }

- P( A

The key to the right use of Bayes is that it can be useful in calculating conditional probabilities: that is, the probability that event A occurs given that event B has occurred. Normally such probabilities are used to forecast whether an event is likely to occur, thus:

Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent years, it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn’t rain, he incorrectly forecasts rain 10% of the time. What is the probability that it will rain on the day of Marie’s wedding?StaTTrek’s solution to Marie’s conundrum looks like this:

“The

sample spaceis defined by two mutually-exclusive events – it rains or it does not rain. Additionally, a third event occurs when the weatherman predicts rain. Notation for these events appears below.

- Event A
_{1}. It rains on Marie’s wedding.- Event A
_{2}. It does not rain on Marie’s wedding.- Event B. The weatherman predicts rain.
In terms of probabilities, we know the following:

- P( A
_{1}) = 5/365 =0.0136985 [It rains 5 days out of the year.]- P( A
_{2}) = 360/365 = 0.9863014 [It does not rain 360 days out of the year.]- P( B | A
_{1}) = 0.9 [When it rains, the weatherman predicts rain 90% of the time.]- P( B | A
_{2}) = 0.1 [When it does not rain, the weatherman predicts rain 10% of the time.]We want to know P( A

_{1}| B ), the probability it will rain on the day of Marie’s wedding, given a forecast for rain by the weatherman. The answer can be determined from Bayes’ theorem, as shown below.

P( A _{1}| B ) =P( A _{1}) P( B | A_{1})

P( A

_{1}) P( B | A_{1}) + P( A_{2}) P( B | A_{2})P( A _{1}| B ) =(0.014)(0.9) / [ (0.014)(0.9) + (0.986)(0.1) ] P( A _{1}| B ) =0.111 Note the somewhat unintuitive result. Even when the weatherman predicts rain, it only rains only about 11% of the time. Despite the weatherman’s gloomy prediction, there is a good chance that Marie will not get rained on at her wedding.

When dealing with conditional probabilities at the loading-end of the formula, we are able to formulate the sample space easily because the “real world conditions” demanded by the formula can be identified, and also have data–predictions– regarding Event B, which is a third event, A1 and A2 being (the required) mutually exclusive events.

So far, you are thinking, this is the kind of thing you would use for weather, rocket launches, roulette tables and divorces since we tend to think of conditional probability as an event that has not happened but can be predicted to happen, or not happen, based on existing, verifiable occurrences. How can it be useful in determining whether events “actually” transpired in the past, that is, when the sample field itself consists of what has already occurred (or not occurred) and when B is the probability of it having happened? Or how it can be useful in dealing with events claimed to be *sui generis* since the real world conditions would lack both precedence and context?

To compensate for this, Carrier makes adjustments to the machinery: historical events are like any other events, only their exclusivity (A or not A) exists in the past rather than at the present time or in the future, like Marie’s wedding. Carrier thinks he is justified in this by making historical uncertainty (i.e., whether an event of the past actually happened) the same species of uncertainty as a condition that applies to the future. To put it crudely: Not knowing whether something *will* happen can be treated in the same way as not knowing whether something *has* happened by jiggering the formula. Managed properly, he is confident that Bayes will sort everything out in short order:

If you treat every probability you assign in the Bayesian equation as if it were a syllogism in an argument and defend each premise as sound (as you would for any other syllogism) Bayes’s theorem will solve all the problems that have left [Gerd] Theissen and others confounded when trying to assess questions of historicity. There is really no other method on the table

since all the historicity criteria so far have been shown to be flawed to the point of being in effect (or in fact) entirely useless. (Carrier, “Bayes Theorem for Beginners,” in Sources of the Jesus Tradition, 107).

What? This is a revolution in thinking? Never mind the obvious problem: If all the *historicity criteria* available have been shown to be “in fact” entirely useless and these are exactly the criteria we need to establish (“treat”) the premises to feed into Bayes, then this condition would make Bayes compeletly useless as well–unless opposite, useful criteria could be shown to exist. Bayes does not *generate* criteria and method; it *depends* on them, just as the solution to Marie’s dilemma depends on real world events, not on prophecy. Obversely, if Bayes is intended to record probability, the soundness of the premises is entirely vulnerable to improbable assumptions that can only poison the outcome–however “unarguable” it is by virtue of having been run through the Carrier version of the Bayes Machine. Moreover, he either means something else when he talks about *historicity criteria* or is saying they exist in some other place. In any event, the criteria must differ from premises they act upon and the conclusion Bayes delivers.

“Fundamentally flawed,” as I noted in a previous post, is the application of Bayes to data where no “real world data and conditions” can be said to apply. It was this rather steep lapse in logic that led a former student of mine, who is now studying pure mathematics at Cambridge to remark,

Is this insistence [Carrier’s] of trying to invoke Bayes’ theorem in such contexts a manifestation of some sort of Math or Physics envy? Or is it due to the fact that forcing mathematics into one’s writings apparently confers on them some form of ‘scientific’ legitimacy?The fact of the matter, as far as I know, and as I thought anyone would realize is that

Bayes’ theorem is a theorem which follows from certain axioms. Its application to any real world situation depends upon how precisely the parameters and values of our theoretical reconstruction of a real world approximate reality. At this stage, however, I find it difficult to see how the heavily feared ‘subjectivity’ can be avoided. Simply put, plug in different values into the theorem and you’ll get a different answer. How does one decide which value to plug in?Secondly, is it compulsory to try to impose some sort of mathematically based methodological uniformity on all fields of rational inquiry? Do there exist good reasons to suppose the the methods commonly used in different areas that have grown over time are somehow fatally flawed if they are not currently open to some form of mathematization?

If this kind of paradigm does somehow manage to gain ascendency, I assume history books will end up being much more full of equations and mathematical assumptions etc. While that will certainly make it harder to read for most (even for someone like me, who is more trained in Mathematics than the average person) I doubt that it would have any real consequence beyond that.”

In fairness to Carrier, however, the use of Bayes is probably not being dictated by logic, or a respect for the purity of mathematics, nor perhaps even because he thinks it can work.

It is simply being drawn (unacknowledged) from the debater’s handbook used by Oxford philosopher Richard Swinburne, who (especially through 2007) was active globally debating the question of God’s existence, under the title “Is there a God?” using Bayes’s Theorem as his mainstay. Not only this, but Swinburne is the editor of the most distinguished collection of essays on Bayes’s Theorem (Oxford, 2002). In case you are interested in outcomes, Swinburne formulates the likelihood of God in relation to one argument for his existence (the cosmological) this way: P (*e* I *h* & *k*) ≥ .50** **The “background knowledge” Swinburne needs to move this from speculation to a real world condition is “the existence [e] over time of a complex physical universe.” In order to form a proposition for debate properly, Swinburne depends on the question “Is There a God,” which gives a clear modality: A and A1.

Unlike Carrier, I believe, I have had the dubious pleasure of having debated Swinburne face to face at Florida State University in 2006. A relatively complete transcript of my opening remarks was posted online in 2010. In case it is not clear, I took the *contra* side, arguing against the proposition.

I knew enough of Swinburne’s work (and enough of his legendary style from graduate students he had mentored at Oxford) to be on guard for his use of Bayes. Unlike Carrier, Swinburne is both a theologian and a specialist in formal logic, whose undergraduate degree was in philosophy, politics and economics. He travels the two worlds with ease and finesse and his most prominent books—*The Coherence of Theism*, *The Existence of God*, and *Faith and Reason-*-are heavy reads.

But he is quite *uncomfortable* with historical argumentation. Historical argumentation is both non-intuitive and probabilistic (in the sense of following the “law of likelihood”); but tends to favor the view that Bayes’s excessive use of “prior possibilities” are subjective and lack probative force. So, when I suggested he could not leap into his Bayesian proofs for God’s existence until he told me what God he was talking about, he seemed confused. When I scolded him that the God he kept referring to sounded suspiciously biblical and fully attributed, he defended himself with, “I mean what most people mean when they say God.” When I retorted that he must therefore mean what most atheists mean when they say there is *not God*, he replied that arguing the atheist point of view was my job, not his. When I said that any God worth arguing about would have to be known through historical documents, the autheticity and epistemological value of which for a debate like this would have to be tested by competent historical research, he became impatient to get back to his formula, which works slowly and cancerously from givens to premises–to the prize: the unarguable conclusion. It seems Swinburne thought the fundamentalist yahoos (not my interpretation) would be so dazzled by the idea of an “unarguable argument” for God’s existence that he would win handily.

Except for those pesky, untended, historical premises. Not to let a proficient of Bayes get past his premises is the sure way to cause him apoplexy, since Bayes is a premise-eating machine. Like any syllogistic process, it cannot burp out its unarguable conclusions otherwise. The result was that in an an overwhelmingly Evangelical-friendly audience of about 500 Floridians, the debate was scored 2 to 1 in my favour: Swinburne lost chiefly because of The Revd. Thomas Bayes.

And this is the trouble Richard Carrier will also need to confront, sooner or later. He will not solve the primary objections to the use of Bayes’s Law by telling people they don’t get it (many do), or that there are no other methods on the table (where did they go to?), or that all existing historicity criteria, to use a more familiar word in the lexicon he uses on his blog, are “fucked.”

It is rationally (still a higher term than logically) impossible to use the existence of the world in which thinking about God takes place as the real-world condition that makes it possible to use cosmology as the real-world condition proving his existence. As Kant complained of Anselm’s ontology, existence is not essence. It is not argument either. The defeater in this case is history: God has one, in the sense that all ideas about God are historically generated and directly susceptible to historical description and analysis.

And he could learn a thing or two from Swinburne’s sad fate, which is adequately summarized in this blog review of the philosopher’s most extensive use of the Theorem in his 2003 book, *The Resurrection of God Incarnate*.

Using Bayesian probability and lashings of highfalutin’ mathematical jargon, Swinburne argues that “it [is] very probable indeed that God became incarnate in Jesus Christ who rose from the dead” (p. 214). His mathematical apologetics for the resurrection boils down to the following argument:

- The probability of God’s existence is one in two (since God either exists or doesn’t exist).
- The probability that God became incarnate is also one in two (since it either happened or it didn’t).
- The evidence for God’s existence is an argument for the resurrection.
- The chance of Christ’s resurrection not being reported by the gospels has a probability of one in 10.
- Considering all these factors together, there is a one in 1,000 chance that the resurrection is not true.
It’s almost impossible to parody this argument (since in order to parody it, you would have to imagine something sillier – a daunting task!). But let me try:

The probably that the moon is made of cheese is one in two (since it is either made of cheese or it isn’t);

the probability that this cheese is Camembert is also one in two (since it’s either camembert or it isn’t); and so on.

At any rate, while Carrier loads his debating machine with still more improbable premises, I am going on the hunt for those missing historicity criteria. They must be here someplace. I do wish children would put things back where they found them.