Bayes’ Theorem and the Historical Truth of Christianity
In his book, Proving History, Bayes’ Theorem and the Quest for the Historical Jesus, atheist activist Richard Carrier contends that reports in Christian scripture can be determined to be false by the application of Bayes’ theorem. He claims that by this method, it can be demonstrated that Jesus of Nazareth is not an historical person but an invented myth.
My past two Catholic Stand posts concerned Raphael Lataster’s judgment of an historical report in scripture. Like Carrier, Lataster is also a proponent of the ‘Christ Myth’ theory. He tries to emulate Carrier’s argument, whom Lataster references, for the understanding of Bayes’ theorem in its application to history. So here I want to take a look at how Carrier attempts to use Bayes theorem to support his contentions.
Any direct answer to Carrier’s challenge to the truth of the Faith, first depends upon understanding Bayes’ theorem. So let’s start there. It’s always good to engage people on their chosen turf.
A Model to Facilitate the Understanding of Bayes’ Theorem
Bayes Theorem is nothing more than a valid algebraic relationship among fractions of a population. It can be visualized using the 48 contiguous United States as a model. The fractions are those of a population. In fact, instead of a population of historical reports, ala Carrier, this model uses a population of people.
Let us first divide the US into two rows, North and South. Then, divide the rows into two columns, West and East. We now have four quadrants, they need not be equal in area. They are definitely not equal in human population. The four quadrants are Northwest, Northeast, Southwest and Southeast. Visualize this two by two division as a map and as a table of data. In the table, as in the map, the North row is on the top, the South on the bottom. The West column is on the left, the East on the right.
If we were to concentrate on only the NW quadrant, Bayes’ theorem permits the calculation of a fraction of the population. The population of the Northwest Quadrant (NW) is divided by the population of the West Column or for short NW/W. We should immediately notice that this is a relationship solely about the Northwest and the West. It says nothing directly about the South row. It also says nothing about the East column.
According to Bayes’ theorem, the fraction, NW/W, equals a numerator, which is the product of two fractions and a denominator of one fraction.
Proving Bayes Theorem
To express these fractions, we must define two other algebraic sums, along with the population of the West column (W). Let the population of the North row be N. Let the total population of the US be T.
The first fraction of the numerator of Bayes’ equation is the population of the Northwest Quadrant divided by the population of the North, or NW/N. The second fraction is the population of the North divided by the total population, or N/T.
The denominator of Bayes’ theorem is the population of the West divided by the total population, or W/T.
Algebraically, we have,
NW/W = ((NW/N) x (N/T)) / (W/T)
To the right of the equal sign, the N’s of the two factors of the numerator cancel out. Also, the T’s on the right hand side cancel out. We are left with the identity NW/W ≡ NW/W which proves the algebraic validity of Bayes’ theorem.
East and West are not linked
In Bayes’ theorem, however, the East population serves solely as the complement of the West population in forming the total population. But it is only the population of the West that is of interest to us. Whether the East population is considered to be one column or a million columns doesn’t change the role of the East. It is still only a complementarity to the West. Similarly, the South population serves solely as the complement of the North population in forming the total population. It doesn’t matter whether the South is considered to be one row or a million rows.
The East population is essentially irrelevant to the calculated Bayesian fraction which solely concerns the West. The fraction calculated by Bayes’ Theorem is the Northwest Quadrant over the West column. The East cannot furnish criteria for evaluating the validity of an internal algebraic relationship within the West.
Application of Bayes’ theorem to the Truth of Historical Reports
There are essentially three defining designations in Bayes’ theorem – the elements of the population, the Top row and the Left column. The bottom row and the right column are simply the nonTop row and the nonLeft column. In the example above, the three designations are humans as the elements, the North as the Top row and the West as the left column.
In an historical context of true reports, the three designations could be (1) reports as the elements, (2) True Reports as the Top row, and (3) Source A reports as the Left column. These are essentially the designations, which Carrier uses. The bottom row would be nonTrue Reports. The right hand column would be historical reports from Sources Other Than A.
Just as the population of the East is irrelevant to an internal algebraic relationship within the population of the West, so too the reports from Sources Other Than A are irrelevant to an internal algebraic relationship of the reports within Source A.
Source A Reports and Sources Other Than A are column designations algebraically no different from X and Y as independent variables. The argument that set Y could determine an algebraic relationship internal to set X, cannot be valid, any more than the reciprocal argument can be valid, namely, that Set X could determine an algebraic relationship internal to Set Y.
Carrier’s Error
Importantly, the algebraic designations of the subsets subjected to the calculation of Bayes’ fraction, including True and nonTrue, must be assigned prior to the calculation, in order to make the calculation possible.
Carrier implies that the algebraically impossible is accomplished by Bayesian algebra. He implies that an algebraic classification internal to the left hand column in a Bayesian scheme of data can be determined by the right hand column. In a Bayesian scheme, classification within each column is independent of classification in the other.
Common jargon, adopted by Carrier, nudges him into error.
Misleading Common Jargon
Common jargon calls the fraction, the top row over the total population, such as N/T, ‘the prior probability’ and Bayes’ fraction, NW/W, the probability ‘consequent’ upon the calculation. The implication is that if we knew that a human belonged to the US population, the probability that he was a Northerner would be N/T. If upon further review, we discovered that this human was a Westerner, the probability of his being a Northerner is NW/W. We could then call N/T the prior probability and NW/W the probability consequent upon the discovery of further review.
In common jargon it is said that the probability has been revised from the prior to the consequent probability. It sounds like the probability is primarily about a specific human and that the probability of his being a Northerner has changed. In fact the two probabilities of being a Northern, N/T and NW/W, did not change. Only our knowledge has changed. Only after the calculation do we know the numerical value of NW/W. Prior to this we knew the numerical value of N/T.
Probability is essentially about subsets and not about individuals. Probability is the fractional concentration of an element in a logical set. It is the ratio of a subset to a set.
Knowledge Changes but not Probability
Comparably, it would be our knowledge that changed and not any probability in the case of Source A and the top row, True Reports (TR). For example, if we knew a report was from the population, the probability that it was a True Report would be TR/T, where T is the total of all reports. If upon further review, we discovered that this report was a Source A report (SA), we could then call its probability, that of the quadrant True Report/Source A with respect to Source A, or TRSA/SA. It sounds like the probability of being a True Report changed from TR/T to TRSA/SA. But it was only our knowledge that changed. By means of the calculation we would know the probability of Source A reports’ being classified as true, whereas prior to the calculation we knew the probability of generic reports’ being classified as true.
Bayes’ fraction tells us nothing about the reliability or the unreliability of the classification of True and nonTrue Reports within Source A. It tells us only the ratio of true reports within Source A to the total of all Source A reports. Richard Carrier’s claim that Bayes’ theorem can be used to test the truth of historical reports is consequently false.
Conclusion
Bayes’ theorem is about fractions involving subsets of already characterized elements. It is not about characterizing elements and thereby placing them into subsets.
At the beginning of his attempt to ‘prove history,’ Carrier makes the false claim that Bayes’ theorem can be used to determine the truth of a statement, thereby placing it into a subset of true statements or into a subset of nontrue statements. Carrier persists in error when he extends that false claim to his quest for the historical Jesus.
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About the Author: Bob Drury
Bob Drury is retired. He has been fascinated with the reasonableness of the Faith since his junior year in high school in the mid20th century for which the religion text was entitled, “Faith and Reason”. That fascination has continued throughout his education in philosophy, math and science. In his essays he hopes to share that fascination with others. Read more at his website, They Have No Wine.

Laurence Charles Ringo