Bradley Bowen and a Lesson on Probability

Jacobus Erasmus, 11 January 2019

Bradley Bowen argues that the resurrection is highly improbable. I wrote a brief response here. Bowen has replied to my response here and here.[1] Now, probability can be a wonderful tool in reasoning if used in the correct way.[2] Unfortunately, there is a lot of confusion about probability. Thus, I think it would be appropriate to discuss Bowen's views and shed some light on the issue.

Let us start by looking at conditional probability. Conditional probability concerns the change in probability when additional relevant information (or background knowledge or assumptions) are taken into consideration. In other words, if A and B are events and P(B) > 0, then the conditional probability of A given the occurrence of B, which we write as P(A|B), is:

P(A|B) = P(A ∩ B) / P(B)

If we wish to know P(A ∩ B), we may cross-multiply the formula and use the following formula:

P(A ∩ B) = P(B) * P(A|B)

We may generalise the formula in the following multiplication rule:

P(φ1 ∩ … ∩ φn) = P(φ1) * P(φn|φ1) * … * P(φn|φ1 ∩ … ∩ φn-1)

As Ernest Adams explains,

This rule says that the probability of a conjunction of arbitrarily many
formulas is the probability of the first conjunct, times the probability
of the second conjunct given the first, times the probability of the
third conjunct given the first and second, and so on up to the
probability of the last conjunct given all of the previous ones.[3]

If we are appealing to independent events (i.e. events that do not affect each other's probability), then we use a simpler formula:

P(A ∩ B) = P(A) * P(B)

Or, to generalise:

P(φ1 ∩ … ∩ φn) = P(φ1) * … * P(φn)

Now, an important question is this: *When should we use the
multiplication rule?* Well, a good rule of thumb is that, if you cannot use
the classical approach, then you should not be using the multiplication
rule. The classical approach uses events, information, or data that have a
numeric aspect that can be described in mathematical terms and used to
calculate probability using counting rules. Such events include, for
example, rolling a dice twice in a row, randomly selecting a ball from a
basket containing ten green and ten blue balls, selecting a card from a
deck three times in a row, and so on. Let me quote from *Probability for
Dummies* as it explains this in the simplest terms:

The classical approach to probability is a mathematical, formula-based
approach. You can use math and counting rules to calculate exact
probabilities in many cases. … Anytime you have a situation where you can
enumerate the possible outcomes and figure their individual probabilities
by using math, you can use the classical approach to getting the
probability of an outcome or series of outcomes from a random process. …
The classical approach doesn't work when you can't describe the possible
individual outcomes and come up with some mathematical way of determining
the probabilities.[4]

This is why probability theorists usually use the multiplication rule solely for (events in) experiments.[5]

Another important question is this: *What types of events or data
should be used in the multiplication rule or as conditioning events?*
The answer, which is obvious, is that we should only use events or data
relevant to the problem trying to be solved. We should not include events
arbitrarily or willy-nilly, nor should we use external events that we have
inferred. As Ruma Falk puts it,

[T]he probability of the target event should be conditioned on the
immediate event given as datum in the problem and not on some inferred
event. … It should be directly defined by the problem's experimental
procedure … The exact method by which we obtained the given data is
crucial in determining our conditioning event.[6]

Now, coming back to Bowen's argument. He argues that 'we must in
general multiply probabilities of individual events to obtain the
probability of a complex event, even when each individual event is
probable, the complex event (or claim) which consists in the conjunction of
those various individual events (or claims) might well be improbable'. He
then claims that the hypothesis that *God raised Jesus from the dead*
assumes or implies the following data/propositions/events:

- (GE) God exists.
- (GPM) God has performed miracles.
- (JEP) Jesus was a Jewish man who existed in Palestine in the first century.
- (JWC) Jesus was crucified in Jerusalem in about 30 CE.
- (DOC) Jesus died on the cross on the same day he was crucified.
- (JAW) Jesus was alive and walking around in Jerusalem about 48 hours after he was crucified.
- (JRD) Jesus rose from the dead.

Bowen then reasons:

The multiplication of probability applies to the claim that Jesus rose
from the dead, (JRD). Suppose that the probability of (JEP) was .8, and
that the probability of (JWC) was .8 given that (JEP) is true (and 0 if
(JEP) is false), and suppose that the probability of (DOC) was .8 given
that (JWC) is true (and 0 if (JWC) is false), and suppose that the
probability of (JAW) was .6 given that (DOC) is true, then the
probability of (JRD) would be approximately: .8 x .8 x .8 x .6 = .3072 or
about three chances in ten. Thus, (JRD) could be improbable, even if the
various individual claims related to it were ALL either probable or very
probable.

Immediately one can see that Bowen is not following the classical approach, since his implied events are not numerical in nature and their probability cannot be calculated using counting rules. Moreover, Bowen does what Falk warns us against: Bowen uses inferred events as the conditioning events. In my response to Bowen, I pointed out that this is not how we calculate the probability of certain hypotheses, especially historical events. If we use Bowen's approach, then many (or most, or perhaps even virtually every) historical event would be highly improbable. Thus, we have numerous other probability formulas, especially the versions of Bayes' Theorem, to calculate the probabilities of historical events or other hypotheses.

At this point, Bowen objects,

Dr. Erasmus complains about my supposed ignorance concerning Bayes'
Theorem, but he inaccurately describes my reasoning by leaving out the
fact that I make use of CONDITIONAL PROBABILITIES, which are crucial to
Bayes' Theorem.

Let us make two points in response. First, my complaint is about how
Bowen tries to apply the multiplication rule to a historical event. This
complaint is valid regardless of whether Bowen is using conditional
probabilities or independent events. Second, Bowen is not following (or
incorrectly using) the multiplication rule and formula, since his
conditioned events do not take into account *all* the relevant prior
events. Moreover, some of his events are independent, such as JEP, JWC,
and DOC. Thus, since Bowen's use of the multiplication rule is incorrect,
and since his making use of conditional probabilities is not the central
issue, I thought it best to represent his argument in a simpler
multiplication equation.

Bowen then criticises my suggestion that we ought to use Bayes' Theorem in order to calculate a hypothesis given some set of evidence:

This tells us NOTHING about the probability of H2 if we don't know
whether C1, C2, or C3 are true! What he has shown is merely that H2 is
highly probable IF we knew for certain that C1, C2, and C3 were true.
This example is irrelevant to the case of the resurrection of Jesus,
where we are not dealing with facts that are known to be true, but are
instead dealing with claims that only have some degree of probability.

Now, of course, Bayes's Theorem is used to calculate the probability of
a hypothesis given the evidence. This is what the theorem is used for. What
I was pointing out is that we should be concerned with any evidence that
raises the probability of Jesus' resurrection and not with propositions
implied by the hypothesis (I should note that, in my other post, I did not
give much thought to the probabilities I assigned in the example, since it
is simply an *example*). Accordingly, my example is relevant to the
discussion.

Bowen then objects to my example of how to use the odds form of Bayes' Theorem:

When he assigned the value of 0.6 to P(H2), he was assigning a moderate
probability to H2, the hypothesis that he was supposed to be using Bayes'
Theorem to SHOW that H2 has a high probability, even when (C1), (C2), and
(C3) have low or moderate probabilities.

Here, as I understand him, Bowen is objecting, not to me, but unintentionally to Bayes' Theorem itself! He takes issue with the fact that one must assign a stand-alone or prior probability to H2 in order to calculate the probability of H2 given the evidence. But this is just how Bayes' Theorem works.

Although Bowen makes several other remarks, I do not see much value in responding to them. What I have said here shows that the resurrection is not necessarily improbable. Let us hope that this post clarifies some of the confusion.

[1] I think that Bowen's keyboard is broken because almost every second word in his posts are UPPERCASE.

[2] Probability is important, for example, not only in philosophy but in Artificial Intelligence (AI). During my postgraduate Honours degree in Information Technology, I had to take numerous lessons on probability for the AI module. Since then, through my programming work, I have used and realised the value of probability calculations in programming, complex systems, and deep learning.

[3] Ernest W. Adams, *A Primer of Probability Logic* (Stanford,
California: CSLI Publications, 1998), 59.

[4] Deborah Rumsey, *Probability for Dummies* (Hoboken, NJ: Wiley
Publishing, 2006), 13–14.

[5] Joseph K. Blitzstein and Jessica Hwang, *Introduction to
Probability*
(Boca Raton, FL: CRC Press, 2015), 8.

[6] Falk, Ruma. ‘Conditional probabilities: insights and difficulties.'
*In Proceedings of the Second International Conference on Teaching
Statistics*, pp. 292-297, 1986, 294.