Betting on horses and the resurrection of Jesus (II)

Versione italiana

Thomas Bayes (1701–1761)

In the last article I used a horse racing / mathematical problem to introduce Bayes' theorem in its odds-based form, ie the one using the ratio of the probability p that an event 'A' occurrs and of the probability 1-p that it does not occur: O (A) = p / (1-p).

At the end of that same article, I also promised a connection between this formulation of Bayes' theorem and the resurrection of Jesus. Obviously the key to this promise is found in the article immediately before, "On the validity of the testimony of the apostles about the resurrection of Jesus", in which I explained why the testimony of the apostles of Jesus about his resurrection is not an evidence strong enough to accept as true the hypothesis that a human being has risen from the dead. Now, with the help of Bayes' theorem, we can calculate how unlikely this hypothesis is.

Bayes' formula

First a small summary. Let O (H) be the odds of the hypothesis 'H' regardless of the occurrence of event 'E', P (E | H) the probability of the event 'E' when the hypothesis 'H' is true, P (E | ¬ H) the probability of the event 'E' when the hypothesis H' is false, and O (H | E) the odds of the hypothesis 'H' when event 'E' is taken into account; then Bayes' theorem is expressed in the following form:

O (H | E) = O (H) * P (E | H) / P (E | ¬ H),

that is the odds of hypothesis 'H' when observing event 'E' are equal to the odds of hypothesis 'H' regardless of the observation of 'E' multiplied by the likelihood ratio, the ratio between the probability to observe the event 'E' when the hypothesis 'H' is true and the probability of the event 'E' when the hypothesis 'H' is false. Through this ratio it is possible to update the odds of a hypothesis following the observation of an event.

The subject of the inquire

Our hypothesis 'H' is "Jesus was truly risen", meaning that Jesus was a human being and that his body returned to life. We have no direct observation of this resurrection, but we observed the event 'E' "the apostles of Jesus were witnesses of the resurrection".

The question we ask ourselves is what is the value of O (H | E), what are the odds that the hypothesis "Jesus is truly risen" is true, taking into account the event 'E' "the apostles of Jesus were witnesses of the resurrection". Applying Bayes' theorem, we find that this quantity depends on three factors:

1. the odds that the hypothesis 'H' is true, regardless of the testimony of the apostles, O (H);
2. the probability that the apostles would have been witnesses of the resurrection if Jesus was really risen, P (E | H);
3. the probability that the apostles would have been witnesses of the resurrection if Jesus was not really risen, P (E | ¬ H).

How many people have lived on Earth?

What is the probability that Jesus rose from the dead, regardless of the testimony of the apostles? Given that this testimony is the only evidence of such an event, it is not unreasonable to think that the resurrection of Jesus has in general the same probability of occurrence of the resurrection of every other human being.

Apart from Jesus, no other human being has ever been risen from the dead, as far as we know, then the probability of rising is less than or equal to one divided by the number of human beings who lived in every time. This number is not known, of course, but there are several estimates and a pretty good guess is around 100 billion human beings, which means that O (H) is approximately 1:100,000,000,000 or 10^-11!

How likely is the testimony of the apostles?

The idea of the apologists is that the will of the apostles to witness to their faith in the resurrection of Jesus even at the cost of life is a very strong evidence for the historicity of the resurrection. To proceed in our account, however, we need to quantify the probability that the apostles have witnessed the resurrection of Jesus, in the case where the resurrection actually took place.

The apologists would like this probability to be very high, because it will increase O (H | E), ie the odds that Jesus rose from the dead when the testimony of the apostles is taken into account. More precisely, what matters is not the absolute value of this probability, but its relationship with the likelihood that the apostles testify if Jesus was not actually risen.

To consider the most favorable case for the apologists, we choose the maximum value for this probability. In theory can not be 100% because it would mean that the resurrection of Jesus invariably involves the testimony of the apostles, and this can not be true, as we must take it into consideration the possibility, however small, that the apostles did not understand the resurrection, or that they decided it was not worth their lives or other events like that. But in order to help the apologists, we arbitrarily choose a value of 100%.

Can you give your life for a false faith?

The last factor to consider is the probability that the apostles testified of the resurrection of Jesus if this event had not occurred. As said, the strength of the thesis of the resurrection depends on the ratio between the previous probability and this one, which the apologists would like to be very low.

Why the apostles would testify to their faith in the resurrection, if it had never occurred? We can not know for sure, but you can suppose, for example, that they were self-deluded, that the shock of the death of Jesus brought them to see visions and these visions had convinced them of the reality of his resurrection.

Whatever the reasons why you would be convinced of the resurrection, it should be noted that this phenomenon is not as uncommon: as written in the previous article, virtually all the Christian martyrs are people who have chosen to pay with their lives for their faith in the resurrection Jesus without being its witnesses.

Therefore an estimate of the probability that people hold their witness for their faith, even at the cost of their lives, should be equal to the ratio between the number of Christians who were witnesses to their faith under threat of death (the martyrs) and that of whole number of Christians who have been confronted with the choice of renounce their faith or die.

How much is this ratio? A few thousand over a few thousand? A few hundred over tens of thousand? It's not easy, but, fortunately, it is not so important to find the correct number.

"Extraordinary claims require extraordinary evidence"

Let's now sum up what we have discovered up to this point. We estimated the odds that Jesus rose from the dead regardless of the testimony of the apostles as 1 in 100 billion, and we have assumed (a little unrealistically) a 100% probability that the apostles would witness with their lives for Jesus' resurrection if this is a real event. In summary:

O (risen Jesus | apostles witnesses) =

O (risen) * P (witnesses | risen) / P (witnesses | not risen) =

10^-11 * 1.0 / P (witnesses | not risen)

It is now clear that there is only one way to increment the possibility of the resurrection of Jesus to a perceptible value: the probability of the testimony of the apostles in the event that Jesus was not raised must be extremely unlikely! To be precise, it must be as unlikely as the resurrection is, or around one in 100 billion, which means that at most one single human being in the entire history should have died to witness something that was false, and it is clear that event is very far from what we know from history and human psychology.

The moral drawn from all of the above is best summed up by a famous quote attributed to Carl Sagan: "Extraordinary claims require extraordinary evidence" means exactly "to make an extremely unlikely event likely, we must have evidence that is equally improbable".

To the misfortune of the apologists, and for the good of reason, Bayes' theorem shows how far the Christians are from establishing the historicity of the resurrection of Jesus.

«Frequentists vs. Bayesians», xkcd.