Bayes Meets God

Bayes theorem is a remarkable bit of mathematics.  It encodes the rational, probabilistic reasoning we all do - even if we don't realise we're doing it.  Suppose I tell you that I spoke to a nice person on the train. In your mind they could be male or female with roughly equal probability.  If I tell you that the individual had long hair you instantly conclude that the person is more likely female than male.  We're doing math and not even realising it.  We consider our hypothesis \( H\) and consider the likelihood of any evidence \(E\) that we may have that supports the hypothesis.  In other words: the probability of the evidence if the hypothesis were true \(Pr(E \mid H ) \).  We also consider the complement of that hypothesis \( \overline{H} \) and the probability of the evidence if the hypothesis were false \(Pr( E \mid \overline{H} ) \).  We weigh this up and draw a conclusion very quickly. Mathematically we're computing:
\[ Pr( H \mid E) = \frac{Pr(H)Pr( E \mid H)}{Pr(H)Pr( E \mid H) + Pr( \overline{H})Pr( E \mid \overline{H})} \] In fact this math tells us why our reasoning works.

As an example, let's test the existence of god.  I may truly not know whether god exists or not so, to be fair, let's say the probability of his (or her or it's) existence before we consider any evidence is \(Pr(H)=0.5 \).  As a consequence of this we also know that the probability that there is no god is \(Pr( \overline{H} )= 1-Pr(H)=0.5 \).  These probabilities are called 'prior' probabilities and assigning the values to these such that all possibilities are equally probable is called a 'non-informative prior.'  Now, I may have a view on what these probabilities could be.  If that's the case I could assign other values to them. This is called an 'informative prior.'  However, if I do, I must be prepared to justify my reasons.

Now I have my prior probabilities set I can turn my mind to the evidence.  If there were a god he would reveal his existence when the faithful pray to him, so I might be particularly interested in whether the prayers of the faithful work.  After all, if there is a god he would answer their prayers right?  It says so in the bible:

"Therefore I say unto you, What things soever ye desire, when ye pray, believe that ye receive them, and ye shall have them." - Mark 11:24

Now, we have all heard of cases where there have been so called miraculous cures from terminal cancer.  It is true that not all such remissions can be explained by science.  The Cancer Council of Victoria state the following on their website.¹

"Spontaneous remission from an apparently terminal cancer is a rare but documented and unexplained phenomenon"

However they also state it may be easier to explain such cases as the science of oncology progresses.  Spontaneous remission in cancer patients has been estimated at 1 in 100,000.²  It does not happen often.  I am specifically interested in those cases where prayer was involved so it might be useful to look at studies on the efficacy of intercessory prayer.  Meta analysis of such studies have concluded that there is "no effect or a potentially small effect" and that "the most methodologically rigorous studies failed to produce significant findings."³  Now, this does not close the door to that possibility of an effect existing due to the potential for type II error in any statistical study.

Remember, at the beginning I said that I want to know how expected the evidence is if my hypothesis were true.  If a true believer prayed for 100,000 people the evidence is saying that 1 will experience a spontaneous remission because little to no effect has been found in studies.  That's shocking!  If god were real the bible tells me I should expect way more than that because prayer is supposed to work.  However, due the potential presence of type II error I'll assume that it's 1 in 50,000 instead.  After all that's doubling the chance of spontaneous remission when compared to studies. This gives me \(Pr(E \mid H ) = 0.00002\) .  It's like saying that god may choose to answer the prayers of the faithful occasionally - a small, but nonetheless significant, effect.

Now let me consider how expected the evidence is if my hypothesis were false.  If no god existed then it is not possible for prayers to be answered by him.  People that were prayed for would have a very small chance of spontaneous remission.  We have the evidence for that too.  The studies show that 1 person in 100,000 will experience spontaneous remission and that the rest won't.  So, how likely is the evidence if there were no god to answer the prayers of the faithful? Very likely - 99,999 people in 100,000 will not experience it so I can safely assert that \(Pr( E \mid \overline{H} ) = .99999\).  I now have all of the numbers needed to do the calculation.

\[ \begin{align*} Pr(H \mid E) &= \frac{Pr(H)Pr( E \mid H)}{Pr(H)Pr( E \mid H) + Pr( \overline{H})Pr( E \mid \overline{H})} \\ &= \frac{0.5 \times{} 0.00002}{0.5 \times{} 0.00002 + 0.5 \times{} 0.99999} \\ &\approx 0.00002 \end{align*} \]  
That number is pretty small right?  What happens if I add additional evidence such as the existence of a heliocentric solar system, the success science has had in explaining the mysteries of the universe, atrocities like slavery or the genocides committed in god's name and so on? What happens if I go through the bible and test each claim?  Clearly the evidence for god gets smaller and smaller and the places he has to hide start to shrink rapidly.  The more evidence like this we have makes the likelihood of a god approach (but not reach - if you want to cling to that) zero.

In order to overcome this it would be required that earth shattering, extraordinary evidence be produced of god's existence.  For now, the evidence is clearly not in his favour.

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1 http://www.cancervic.org.au/about/media-releases/2010-media-releases/october-2010/cancer-prayer.html^↩
2 Hobohm U (October 2001). "Fever and cancer in perspective". Cancer Immunol. Immunother. 50 (8): 391–6. PMID 11726133^↩
3 http://en.wikipedia.org/wiki/Studies_on_intercessory_prayer^↩