St. Andrew’s Episcopal Church
Faith, Probability and Bayesian Inference
Paul G. Bamberg
Senior Lecturer on Mathematics, Harvard University

The Bayesian Bible

Here are four examples of how the Bible would have to read in order for straightforward Bayesian analysis in terms of conditional probability to be directly applicable.  Each is based on a classic example from probability theory.

At the second class, on December 15, you will learn how to analyze the problems posed in the apocryphal texts below.  At the third class, on December 22, we will try to decide whether or not this type of reasoning, or a less restrictive version based on the concept of likelihood, is useful in analyzing questions of religious belief and interpretation of the Bible.

Words in italic bold type are from the Bible, Revised Standard Version.  Everything else is a fabrication of the author.

Using an “effect” to assess the probability of a “cause”

Acts 16
1.	And he (Paul) came also to Derbe and to Lystra.  A disciple was there, named Timothy, the son of a Jewish woman who was a believer, but his father was a Greek.
2.	Now the inhabitants of Lystra numbered 2000 Jews and 8000 Greeks. Among the Jews, half were believers, but among the Greeks, only one of every four was yet a believer.
3.	And Paul commanded Timothy to go out into the market of the city, to assemble a group of believers, and to choose one of them by lot to be brought to Paul.
4.	And Timothy did as he was commanded, and returned to Paul with a believer, saying “My chosen believer waits outside, but I know not whether he is a Jew or a Greek.”
5.	Paul replied, “Of believers thus chosen, two of three will be Greeks.”
6.	Timothy replied, “How can this be, seeing that the fraction of believers is higher among the Jews?”
7.	And Paul said, “Bring the man to me, and we shall learn.”  And indeed the man was a Greek, named Nicholas.  Both he and Timothy were astounded at Paul’s wisdom, and resolved to accompany him on his travels.

Question: How did Paul calculate the probability of 2/3 that the unseen believer was a Greek? 

A version of the classic “Monty Hall” problem

Mark 9
4.    And there appeared with them Elijah and Moses; and they were talking to Jesus.
5.    And Peter said to Jesus, “Master, it is well that we are here; let us make three booths, one for you and one for Moses and one for Elijah.”
6.	And the booths were made, and a cloud overshadowed them, and Jesus occupied one booth, and Moses another, and Elijah the remaining one. Peter knew not in which booth Jesus was to be found, and cried in despair, “Where is my Master?”
7.	And a voice came from the cloud, saying, “I have created you with free will; therefore choose.” Peter did as he was commanded and chose a booth, but he dared not open the door.
8.	And the voice spoke again, saying “Those who seek the Master must open the door themselves, but I will show you the most ancient prophet.” And the door of one of the booths that Peter had not chosen was flung open, and a prophet appeared, and the voice said, “This is not my beloved Son, although you should listen to him.”
9.	Peter perceived that there were two booths whose doors were yet unopened, and he cried aloud, “I chose freely and without partiality, but I now perceive that my choice was unwise.” And he boldly threw open the door of the third booth, neither the one that he had chosen nor the one that God had opened.
10.	 Jesus emerged and said, “Peter, you have chosen well.  For nothing is certain in this world, but those who interpret the word of God wisely will open the right door more often than not.”

Questions: 
Peter correctly concluded that the probability that Jesus was in the third booth was 2/3.  How did he reach this conclusion?
What is the probability that the prophet in verse 8 was Moses rather than Elijah?

Making inferences based on statements from untrustworthy witnesses

1 Kings 10
1.	Now when the queen of Sheba heard of the fame of Solomon, concerning the name of the Lord, she came to test him with hard questions.
2.	She came to Jerusalem with a very great retinue, with camels bearing spices, and very much gold, and precious stones, and when she came to Solomon, she told him all that was on her mind.
3.	And she said to Solomon, “All these spices and gold and precious stones I will exchange for one true prophet.”
4.	Now there were in Jerusalem many prophets: true prophets who spoke nine parts truth and one part falsehood, and false prophets who spoke truth and falsehood in equal measure.  But only Solomon himself was without any falsehood.
5.	And Solomon sent his servant into the city to gather three prophets from which the queen might choose.  
6.	The servant returned with three prophets and said to Solomon, “Master, two of these are true prophets, but one is a false prophet, and I have forgotten which.”
7.	Now when the queen heard these words, she was dismayed and said to Solomon, “I cannot return to my land fearing that there is one chance in three that I have chosen a false prophet.  One chance in ten would be acceptable.  Therefore command your servant to find more true prophets before I make my choice.”
8.	“My servant is weary and deserves to rest,” replied Solomon.  “But I will question one of the prophets to learn more about who is a true prophet.”
9.	“How can that be,” asked the queen, “since any of these prophets might speak falsehood in response to your question?”
10.	Solomon then addressed the first prophet, saying, “Is the second prophet a true prophet?” 
11.	 The first prophet replied, saying, “Master, he is not.”
12.	Solomon took the third prophet by the hand, saying, “I know that there remains one chance in fifteen that you are a false prophet.  Nonetheless, you are likely to bring credit to Jerusalem as you serve your new mistress.”
13.	And King Solomon gave to the queen of Sheba all that she desired, what ever she asked besides what was given to her by the bounty of King Solomon. So she turned and went back to her own land, with her servants.

Questions:
How did Solomon deduce that the probability that the third prophet was false was only 1/15?
What would Solomon have done if the first prophet had answered “Yes” to his question?

Choosing between models with an infinite number of possible outcomes.

Revelation 22
10.  And he said to me,  “Do not seal up the words of the prophecy of this book, for the time is near.”
11.	“But no one shall know the year of my return except my Father. For I have prepared a golden coin, with the image of the head of a horse on one side, and if I toss the coin and that side is uppermost I shall return in that year.”
12.	“Likewise, I have prepared a cube, with each side faced with precious stone: one jasper, one carnelian, one marble, one obsidian, one alabaster, and one turquoise. And if I toss the cube and the side with carnelian is uppermost I shall return in that year.”
13.	After the passing of two millennia, my Father will choose at random either the coin or the cube for me and will cast the other into the Pit.  
14.	At each new year I shall toss the coin or the cube that my Father has given to me.  If the horse or the carnelian appears, I shall return, else I shall bide my time for yet another year.  You must be ready, for the Son of man is coming at an unexpected hour.
15.	And I said to him, “Master, you say that none but the Father knows the year of your return.  Yet I perceive that in the year following the second millenium there is one chance in three that you will return.  
16	If after two years you have not returned, there are 25 chances of 34 that your Father has chosen the cube and has cast the coin into the Pit. Therefore the faithful will become less watchful.
17	He said to me, “John, you perceive what it is unwise for man to know, Therefore I command you to expunge from this book the words that you have just written, and to replace them with the statement that I am the Alpha and Omega.”
18	I warn everyone who hears the words of prophecy of this book: if anyone adds to them, God will add to him the plagues described in this book.

Questions:
How does John deduce that the probability of return in 2001 is 1/3?
How does John deduce that that conditional probability that the die was selected rises from ½ to 25/34 after two years?
If Jesus does not return in 2001, what is the probability (according to this model) that he will return in 2002?