Bayesian Probability Theorem

Often heard in response to the arguments of historical apologetics, such as the claim that God raised Jesus of Nazareth from the dead, is the axiom ‘extraordinary claims require extraordinary evidence.’ People who use this line must be unaware this has been soundly refuted in current philosophical thought, or else persuaded to use irrational principles to satisfy the requirements of their ideological allegiances. After all, there are hundreds of extraordinary claims you come across each day, and yet have no trouble believing.

Consider the lottery reported last night on television as one such event. The chances of winning, or indeed any random sequence of numbers, is extraordinarily improbable, yet if it is true that extraordinary claims require extraordinary evidence, you should never believe it happened. Weighing the probability of the extraordinary event will swamp the reliability of the witnesses every time so that you should never believe it. Even if the programs reporting is 99.9% accurate.

This kind of thinking is really a popular hang-over of Hume’s problem with miracles, which has been thoroughly refuted. John Earman, the agnostic philosopher wrote the book called Hume’s Abject Failure, in which he argues as commonsensical as this principle sounds, it is demonstrably false. The problem that probability theorists have worked on is how one can establish highly improbable events. They realised that you also have to consider the probability that if the reported event would not have occurred that the event would have been reported as it is. 

For instance, what is the probability that the sequence of numbers reported as the lottery result, would have been reported had those numbers not been the correct result. In the case of the resurrection, what is the probability that if Jesus of Nazareth did not rise from the dead we would have the evidences of the empty tomb, the post-mortem appearances and the origin of the disciples belief, et cetera?

Thus an elegant way to assess highly improbable events was developed. The probability for hypothesis (H) on the given evidence (E) with respect to the general background knowledge (B), called Bayes’s theorem. 

bayes

How this works is you plug in values of >.5 for positive probability or <.5 for some negative probability. As the result moves towards 1 it is more likely and towards 0 it is less. In the numerator we have the intrinsic probability of H multiplied by H’s explanatory power, Pr(E/H). The intrinsic probability of H is the conditional probability of H relative to the background knowledge (B). The Pr(E/H) is the rational expectation of E given H is the case, again relative to the background knowledge (B). The background knowledge in both cases is tactically assumed. In the denominator the above product is added to the product of the intrinsic probability and explanatory power of the denial of H. If this latter product is 0 then the numerator and denominator are the same and yield a ratio equal to 1, meaning 100% probability.

Hume failed to appreciate the probability calculus which entails not only the general background knowledge of the way the world is, but also the probability that we should expect the given evidence had the proposed event not occurred. So it turns out that it could very well be the case that an extraordinary event would not require extraordinary evidence, if the evidence is highly unlikely to occur had the event not taken place. He confuses miraculousness with probability and infrequency with implausibility. That’s one reason why Hume’s argument against miracles is entirely fallacious. 

Richard Swinburne, the philosopher of science from Oxford University, after plugging in the values, gives the probability of Jesus’ resurrection from the dead as 97%. Now I’m not sure how he arrived at the values he plugged in, so I wouldn’t necessarily use Bayes’s Theorem for an analysis of philosophical hypotheses such as God raised Jesus from the dead. But I think enough has been said to show that extraordinary claims do not require extraordinary evidence.

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  1. gary
    gary says:

    When a Christian starts using complex mathematical formulas and philosophical theories to defend his belief in first century corpse reanimation-transformation (aka:
    resurrections)…I yawn.

    I yawn because it is soooo silly.

    I know for a fact that if a Muslim attempted to use these same ploys to defend the veracity of Islam’s claim that Mohammad flew to heaven on a winged horse, the very same Christians would snicker and hand-wave away these arguments without giving them a second thought, believing that these tactics are nothing more than an obvious, desperate attempt to dress up a superstition as believable reality.

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