MY UNDERSTANDING OF PROBABILITY
Dr. Jacobus Erasmus has raised an objection to one of my posts on the resurrection.
Before presenting his objection he takes a swipe at my credibility:
…Bowen’s argument is an example of what happens when a blogger who is untrained in probabilistic logic tries their hand at probability.
…Bowen does not seem to be aware of Bayes’ Theorem; it appears that he has come up with his own idea of how probabilities should be calculated.
I’m not sure what Dr. Erasmus knows about my education, because he provides NO FACTS or evidence about my education. It is not correct to say that I am “untrained in probabilistic logic”. I took multiple courses in logic and critical thinking as an undergraduate student of philosophy, and I took multiple courses in logic and was a teaching assistant for multiple courses in logic and critical thinking as a graduate student of philosophy. I learned some basic probabilistic logic in these courses both as an undergraduate student and as a graduate student.
I did not learn about Bayes’ Theorem in the logic and critical thinking classes that I took nor in the logic and critical thinking courses that I helped teach. So, that is a potential weakness in my educational background. However, I have been studying Richard Swinburne’s book The Existence of God for many years, and that has required that I develop a basic understanding of Bayes’ Theorem. Most of what I have learned about Bayes’ Theorem comes from Swinburne, who is an expert on this subject. Although I have not had any courses that included Bayes’ Theorem, I have learned about this theorem from a qualified expert.
SOME DOUBTS ABOUT DR. ERASMUS’ UNDERSTANDING OF PROBABILITY
Since Dr. Erasmus questions my credibility in terms of my educational background, I will return the favor. His educational background is in Information Technology (an undergraduate degree) and Philosophy (a PhD). So far as I know, courses in Probability are NOT required for either Information Technology nor for Philosophy degrees. So, it is not clear to me that Dr. Erasmus has had ANY classes in Probability. I suspect that he has had some classes in logic, but logic classes don’t necessarily cover Probability calculation. Maybe Dr. Erasmus took a Probability class or two in order to fulfill a math or logic requirement. I don’t know. But his degrees don’t imply that he has ANY background in probability calculation.
There are a few obvious problems with Dr. Erasmus’ short post that indicate to me that he does NOT understand probability well.
First, he provides NO EXPLANATIONS in his post. He neither EXPLAINS my alleged error, nor EXPLAINS his own example of a probability calculation. If he really understood probability, then I would expect him to clearly EXPLAIN both points, so the absence of any such explanations suggests to me that he doesn’t really understand what he is talking about.
Second, he contradicts himself in the counterexample that he provides. On the one hand, he assigns an estimated probability of 0.6 to the claim that The butler is a murderer. But then he immediately turns around and calculates a probability of 0.9 that the butler murdered Jones. Those two probabilities CANNOT BOTH BE CORRECT. A third grader could see that! But not Dr. Erasmus.
If the probability that the butler murdered Jones was truly 0.9, then the probability that The butler is a murderer must be AT LEAST 0.9; it can’t be less than 0.9. There is some chance that the butler murdered somebody else besides Jones, so the probability that The butler is a murderer must be 0.9 plus the probability that the butler murdered someone else besides Jones. Dr. Erasmus contradicts himself in the space of just a few paragraphs while presenting his counterexample to my reasoning. I am not impressed by such sloppy thinking.
Third, one of the pieces of evidence in Dr. Erasmus’ counterexample makes his example inappropriate:
The butler was the only other person in the house when Jones died.
This bit of evidence all by itself makes it highly probable that the butler murdered Jones. The counterexample involves the murder of Jones by means of someone hitting him in the head with a brick. If the butler was “the only other person in the house when Jones died”, then it would be nearly impossible for anyone other than the butler to have committed the murder of Jones. This one bit of evidence makes the other evidence largely irrelevant. Given this one bit of evidence, it is already determined that it is highly probable that the butler murdered Jones. But Dr. Erasmus fails to see this obvious point.
Furthermore, this makes the supposed counterexample a poor one, since the probability calculation concerning the resurrection of Jesus does NOT include such a bit of evidence that all by itself could settle the issue, at least not in support of the hypothesis. Many of the claims that I consider are necessary conditions of the hypothesis “God raised Jesus from the dead”. The falsehood of a necessary condition would thus immediately establish with certainty the falsehood of the hypothesis. But none of the claims I consider would all by itself show the hypothesis to be true or highly probable.
Fourth, 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. So, Dr. Erasmus mischaracterizes my reasoning in a way that divorces my reasoning from the logic of Bayes’ Theorem. It is very clear that I am making use of CONDITIONAL PROBABILITIES when I talk about multiplication of probabilities in the case of assessing the hypothesis that “God raised Jesus from the dead”:
(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.
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.
[an excerpt from my post that Dr. Erasmus is criticizing]
If Dr. Erasmus is familiar with probability calculations, then he would know that the expression
…the probability of (JWC) was .8 given that (JEP) is true…
is a reference to CONDITIONAL PROBABILITY. But there is no hint in Dr. Erasmus’ post that he is aware that I was making use of CONDITIONAL PROBABILITIES.
So, either he FAILED to notice this obvious and important element of my reasoning, and thus shows himself to be ignorant about probability calculations, or else he DID notice this obvious and important element of my reasoning, but he dishonestly suppressed this fact in order to make me appear to be ignorant about probability calculations. Bayes’ Theorem is derived from a basic principle of CONDITIONAL PROBABILITY.
Fifth, Dr. Erasmus appears to infer an UNCONDITIONAL PROBABILITY, when the instance of Bayes’ Theorem that he spells out clearly only establishes a CONDITIONAL PROBABILITY:
In this case, the odds in favour of H2 is about 10:1 (ten to one), which converts to a probability of 0.9 (or 90%) for H2.
He appears to infer that the unconditional probability of H2 is 0.9, but that is NOT what his instance of Bayes’ Theorem shows. The left side of his instance of the equation contains CONDITIONAL PROBABILITIES:
P(H2 | C1 & C2 & C3) / P(~H2 | C1 & C2 & C3)
So, what Dr. Erasmus is calculating is the relative probability of H2 vs. not-H2, given that C1 and C2 and C3 are true.
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.
It might be the case that Dr. Erasmus has more “training” or education than I do about Bayes’ Theorem, but his degrees don’t show that to be the case, and the various problems with his post (that I have pointed out above) suggest to me that he does NOT have a good understanding of probability.
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