Bayesian Reasoning in Religious Studies

Dr. Raphael Lataster is a university lecturer in religious studies and a strong advocate of Bayesian reasoning. His work’s central theme is the inauthenticity of much of the subject matter of the Judeo-Christian scriptures, including the historicity of Jesus of Nazareth. He identifies Bayes’ Theorem as the most useful method of deciding which of several explanations, within any topic, is most probably true (See Section 3 and the video at the 45th  minute).

Lataster’s Use of Bayesian Reasoning

Illustrating the employment of Bayes’ Theorem in determining the authenticity of an historical claim in a “Case Study: The Death of Herod Agrippa”, Section 4, Lataster concludes,

The theory that Herod died of natural causes, ~h, includes the explanation, ‘the claim was simply fabricated’ … Given that such evidence works well for the alternative theory of fabrication, there is no extraordinary evidence that overcomes the inherently low prior probability. As this case study demonstrates, Bayesian reasoning is formally and mathematically valid, even if accurate calculations are not done.

Lataster had already established that the “prior” probability is infinitely small:

The inherent probability of the theory (without yet considering the available evidence, such as the reference in Acts), P(h|b), is infinitely small.

The Relationship of Bayesian Reasoning to the Theorem

Bayes’ Theorem permits the calculation of a probability, P, as equal to a Factor times the Prior probability.

P = Factor × Prior

Mathematically, the product of any positive number times zero equals zero.

However, Lataster’s argument is not that Prior equals zero, but that it is infinitely small, in the case of evaluating the probability of the historicity of the report in Acts of Herod’s being struck down by an angel. He claims that employing Bayesian reasoning without knowing the numerical value of Factor, one can conclude that P is infinitely small; i.e., nearly zero. Thus, Herod’s being struck down by an angel, as reported in Acts, is historically inauthentic because its probability, as a report in Acts, is nearly zero.

A Clarifying Illustration of Bayes’ Theorem

To aid in understanding Bayes’ Theorem and Lataster’s Bayesian reasoning in the evaluation of the historical data cited by him, we will numerically consider a modern data set from 2010.

Before we look at that modern data, let us get a clearer idea of the meaning of the Prior and the meaning of P, the probability calculated using Bayes’ Theorem.

The Prior is the probability that any generic element of a set belongs to subset X prior to knowing that the element belongs to the specific subset Y. An example of a Prior probability would be the probability that any ranch horse in Montana was an Appaloosa (subset X, i.e. all the appaloosas on ranches in Montana), prior to knowing that the horse belonged to the ranch, Barrel H (specific subset Y, i.e. all the horses on the Barrell H ranch). Notice that the set, of which X and Y are subsets, is all the ranch horses of Montana.

The probability calculated by Bayes’ Theorem is the specific probability, in this example, the probability that a horse of specific ranch Y (the Barrel H) is of breed X (Appaloosa).

The Prior probability of an appaloosa applies to ranches generically (the ranches of Montana). The calculated probability of an appaloosa applies to a specific ranch (the Barrel H ranch).

Bayes’ Theorem Using a Modern Data Set

For illustration, the generic set is the set of world residents in 2010. The X subset comprises the residents of Ismay, Montana. The Y subset comprises the residents of Custer County, Montana, in which the town of Ismay is located. (Appaloosas are Ismayans. Custer County is the Barrel H.)

The population of the world in 2010 was estimated at 6.8 billion people. The U.S. Census of 2010 reports the population of the town of Ismay as 19. The Prior probability that any generic resident of the world was a resident of Ismay in 2010 would be 19 divided by 6.8 billion or 2.8 × 10-9, i.e., a probability of a little less than three per billion.

Let it also be given that Factor equals 5.7 × 10-5. This is the Factor, which multiplied by Prior yields the probability that a 2010 resident of Custer County was a resident of Ismay. The calculated probability by Bayes’ Theorem is:

P = (5.7 × 10-5) × (2.8 × 10-9) = 1.6 × 10-3 or 1.6 per thousand

However, employing Lataster’s Bayesian reasoning, we did not need to know the numerical value of Factor. By just knowing that the Prior of 2.8 per billion was close to zero, we could conclude that P was likewise close to zero. It would be improbable for a resident of County Custer in 2010 to claim to have been a resident of Ismay. Such a claim would be historically inauthentic because the arithmetic product of any positive number times 2.8 per billion is nearly zero.

Bayesian Reasoning: Not Depending upon Bayes’ Theorem for Precision

Notice that Bayesian reasoning did not depend upon Bayes’ Theorem and a numerical value of Factor. According to Bayesian reasoning, the fact that the Prior of 2.8 per billion is nearly zero, mathematically necessitates that P be nearly zero.

However, setting such Bayesian reasoning aside, did we have to depend upon Bayes’ Theorem to calculate P? We did if all we knew were the numerical values of the Prior and the Factor given above. We did not if we knew from the U.S. census of 2010 that the population of Ismay was 19 and the population of Custer County was 11,699. We wouldn’t have to know that the Prior, i.e. the world probability of being a resident of Ismay in 2010, was 2.8 per billion. We could simply divide 19 by 11,699, which is 1.6 per thousand. This is the probability that a resident of Custer County in 2010 was a resident of Ismay.

Bayesian Reasoning: Not Depending on Bayes’ Theorem for Precision by Lataster

Would it be possible to employ the same tactic to corroborate or discredit Lataster’s conclusion from Bayesian reasoning that the probability of the cause of an individual’s death by an angel within Acts is nearly zero because the Prior is nearly zero?

In Lataster’s Case Study, the generic set is deaths of all individuals to date from whatever cause. The X subset is the deaths of all individuals caused by being struck by an angel. The Y subset is the deaths of all individuals reported in Acts from whatever cause. (Appaloosas are deaths by an angel. Acts is the Barrel H.)

It is estimated that one hundred billion individuals have died over time. If the cause of death of one individual was death by an angel the generic probability, i.e. the Prior, would be 10-11 or one in two hundred billion. Along with Lataster, we could conclude, based on Bayesian reasoning, that the probability of the cause of death of an individual by an angel, specifically within Acts, must also be nearly zero.

In other words, of the reports of individual deaths in Acts, the probability that any one death was therein attributed to being struck by an angel is nearly zero. Further, that report, according to Bayesian reasoning, is inauthentic. It is a fabrication by the author of Acts because it is nearly zero when compared to the very large total of deaths, which the author of Acts relates as occurring within the context and timeframe of Acts.

But, suppose we actually counted the number of deaths reported in Acts within its context and timeframe. We could then calculate the probability of the cause of death by an angel specifically within Acts without recourse to Bayes’ Theorem or Bayesian reasoning.

I counted six named deaths (Ananias, Sapphira, Stephan, Tabitha, James, and Herod). I took four as the count of those unnamed guards executed by order of Herod for letting Peter escape. The probability of a report of an angel as the cause of death in Acts, which is the specific subset, would then be 1/10, one in ten.

Although one in two hundred billion is nearly zero, one in ten is not. There is probably something wrong with Lataster’s Bayesian Reasoning.

Such Bayesian reasoning is also inconsistent with Bayes’ Theorem. The factor would be 1010 and the calculated probability specific to Acts would be 1/10.

Conclusion

In mathematics, the product of any positive number times zero is zero. A non-corollary of this is the product of any positive number times a number nearly zero is nearly zero. This non-corollary is false.

We must reject the validity of the non-corollary and with it the Bayesian reasoning of Lataster’s argument as a means of historical criticism in both religious and non-religious studies. Also, Bayes’ Theorem concerns the ratios of data counts. Lataster invokes Bayesian reasoning for the purpose of historical criticism, which concerns datum content.

Note: For a more detailed explanation of Bayes’ Theorem see here and here. Subsets in Bayes’ Theorem are defined as complementary, with their data content taken for granted. For a demonstration that Bayes’ Theorem cannot be the basis of a critique of datum content see Bayes, Baseball, and Bowling.

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4 thoughts on “Bayesian Reasoning in Religious Studies”

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