Testing a Hypothesis about the Supernatural

[polymath257]

I agree that it exists or it doesn't. And no matter how much we choose to believe or not believe in it, it makes it no more or less real.
But, if we have no way of testing for the supernatural, how can we even determine that it exists in the first place?
If 'Event A' happens, and it's claimed to be supernatural, there must be something about that event that gives you reason to believe that it is "supernatural" and not just "unknown". Supernatural is not synonymous with unknown, which is what a lot of people seem to be doing.

Okay, but context is important. I said earlier that when discussing Jesus' miracles, the context that strengthens the claim might include:

1. Timing
2. Illustrating a particular point. 
3. Reinforce teachings with some authority. Example feeding 5000, Matt 9:35
4. So that people might believe (specifically stated). Example Lazarus (John 11)
5. Reward for faith.
6. Theologically significant. example virgin birth, baptism, tearing of the veil in the temple, resurrection.

So let's stick with the example I have above. So as not to get sidetracked on a debate about the NT, let's just say for the sake of this discussion you were present and you knew the man to be paralyzed. 

Luke 5:17 On one of the days while Jesus was teaching, some proud religious law-keepers and teachers of the Law were sitting by Him. They had come from every town in the countries of Galilee and Judea and from Jerusalem. The power of the Lord was there to heal them. 18 Some men took a man who was not able to move his body to Jesus. He was carried on a bed. They looked for a way to take the man into the house where Jesus was. 19 But they could not find a way to take him in because of so many people. They made a hole in the roof over where Jesus stood. Then they let the bed with the sick man on it down before Jesus. 20 When Jesus saw their faith, He said to the man, “Friend, your sins are forgiven.”

21 The teachers of the Law and the proud religious law-keepers thought to themselves, “Who is this Man Who speaks as if He is God? Who can forgive sins but God only?” 22 Jesus knew what they were thinking. He said to them, “Why do you think this way in your hearts? 23 Which is easier to say, ‘Your sins are forgiven,’ or, ‘Get up and walk’?

24 “So that you may know the Son of Man has the right and the power on earth to forgive sins,” He said to the man who could not move his body, “I say to you, get up. Take your bed and go to your home.” 25 At once the sick man got up in front of them. He took his bed and went to his home thanking God. 26 All those who were there were surprised and gave thanks to God, saying, “We have seen very special things today.”

Present in the series of events is 1, 2, 3, 5, and 6. That's a lot of context. 

Now, using Bayes Theorem and especially Bayesian Inference, we can examine the probability of seeing the paralyzed man walk given the overall context. 



R = A Miracle Having Happened (the man walks due to supernatural causes)
B = Background information (the supernatural exists)
E = Evidence (paralyzed man walking in the context of being commanded to for the reasons mentioned)

The way you read this is 
Pr="The probability of" 
| = "given"
& = "and"

So the probability of a Miracle Having Happened given the Evidence and The Supernatural Exists OVER the probability of a Miracle Having NOT Happened  given the Evidence and The Supernatural Exists
=
The probability of Miracle Having Happened given The Supernatural Exists OVER the probability of Miracle Having NOT Happened given the The Supernatural Exists
X 
The probability of seeing the Evidence given a Miracle Having Happened and The Supernatural Exists OVER the probability of seeing the Evidence given a Miracle NOT Having Happened and The Supernatural Exists

Notice this last part of the equation. It is the probability of seeing the evidence given no miracle, no supernatural. A low value here significantly increase the overall probability of a miracle having happened.

Well, there are several issues here. First of all, the assumption B is in dispute. So you really want P(R|E), not P(R|E&B). In effect, we can eliminate B from all of this because it is irrelevant to the calculation.

A better analysis is to consider the following:
R= a miracle occurred
S= a story was told about a miracle


What we really want is the probability that a miracle occurred given that we have the story of such a miracle, P(R|S). We can argue separate situations of P(E|S), the probability that a paralyzed man walked given that a story was told that such happened. But that isn't necessary for this calculation.

Now, we have the following (I'm not doing a picture, but it is essentially your equation again):

P(R|S)/P(~R|S) = [P(S|R) P( R)]/[P(S|~R)P(~R)]

In this, the left side is the ratio between the probability that a miracle occurred given that we have a story and the probability that it did NOT occur given the story.

Now, we can agree that P(S|R), the probability of a story given a miracle, is high. But we can also agree that P®, the probability of a miracle is low (whether or not there is a supernatural, miracles have low probability). So, the top has size roughly P( R), which is low.

Now, in the denominator, P(S|~R), the probability of a story when there is no miracle is moderately high. This was a superstitious culture and stories like this were common (not just for Jesus, mind you). And, finally, given that P( R) is low, P(~R) is high. This makes the denominator on the left moderately high.

But the *ratio* between a low value (numerator) and a moderately high value (denominator) is *low*. This means P(R|S) is smaller than P(~R|S), in other words, the probability of a miracle given the story is smaller than the probability of no miracle given the story.

In other words, by your method, we see that it is unlikely that a miracle actually occurred.