Critique of a Bayesian Case Study of Herod’s Death by an Angel

Raphael Lataster is a teaching fellow in the studies of religion at the University of Sydney. Lataster and Trent Horn, a Catholic apologist, debated one another on the topic Does God Exist? in May, 2016 at the University. At the 45^th minute of the debate, Lataster chides Horn for failing to formulate probabilistic arguments in favor of theism, factoring in the alternatives, and doing the calculations which weigh out the probabilities. By this Lataster meant that Horn failed to present an argument for theism based on Bayes’ theorem.

This essay is a critique of Lataster’s own track record in applying Bayes’ theorem in what he calls, “A Case Study: The Death of Herod Agrippa”. This essay is the first of two. Next month I will critique Lataster’s argument without regard to Bayes’ theorem.

What is Bayes’ Theorem?

It is an equation which permits the algebraic calculation of a fraction of a population, given other fractions of the population. Bayes’ theorem does not apply to just any population, but to a population where each member has an ID of just two symbols. One symbol of the ID is from one category of symbols and the second symbol is from a second category of symbols.

This means the population can be presented in a table as a rectangular array of cells. The population may be expressed as elements of data or as percentages. The tabular array could be of any size. However, it is often a table of two rows by two columns, thereby forming four cells. Each cell is identified by its row and its column. Lataster’s case study of Herod’s death is an analysis of a two by two table.

Bayes’ theorem is the algebraic expression for a fraction. It is the fraction which is the ratio of the number of members in a cell divided by the sum of the column to which the cell belongs. However, only the ratio is calculated by Bayes’ theorem, not the numerator and the denominator, individually.

An Example of a Bayesian Population

The column headings or markers are often considered to be observable. As an illustration let a population of beef steers be of two different breeds, Herefords, or H and Black Angus, or B. The headings of the two columns of our population table would be H and B, which are considered observable ID markers. Even I can tell a Hereford from a Black Angus. Let the population of steers belong to two ranches X and Y. Let each steer be branded accordingly with an X or Y, but let the brand be detected only with some difficulty.

Let us illustrate this as numbers of 100 head of steers and as percentages of the total population

As noted above, Bayes’ theorem is an algebraic expression which equals the fraction of the number in a cell divided by the sum of the column of that cell.

For illustration, let our cell be that of row X and column H. That cell is the number of Herefords branded X, which is 50. Let our notation for this be Cell(X,H). The fraction, which can be calculated using Bayes’ theorem is Cell(X,H) / Sum Column H. That is the fraction of Herefords in this population, which are branded X.

The simplest calculation of this fraction would be from the table. It would be 50/75 = 66.7% or, in percentages, 34.5% / 51.7% = 66.7%. But suppose we do not have the information in the table.

Bayes theorem is,

Cell(X,H) / (Sum column H) =

((Sum row X) / (Sum Table) * (Cell(X,H) / (Sum Row X)) / (Sum Column H) / (Sum Table) =

(Fraction One * Fraction Two) / Fraction Three

Without having the information in the table, we could calculate the fraction of Herefords, which are branded X, if we knew the values of Fractions One, Two and Three, respectively as 55.2%, 62.5% and 51.7%. Using Bayes’ theorem, the fraction, Cell(X,A) / Sum Column H, is 66.7%.

The calculation is the fraction of Herefords of this population branded X. It is 66.7%. The calculation is based on knowledge prior to the calculation, namely the values of three fractions. However, in Bayesian jargon only one of these fractions is called ‘the prior’.

The usefulness of the calculated ratio may be described thusly:

A steer is initially identified as a member of the population. Therefore, we know the probability that it is branded X is 55.2%, prior to further review. ‘The prior’ is 55.2%. Upon further review, the generic steer is identified to be a Hereford. Consequent upon or posterior to further review, we know the probability that this steer is branded X is 66.7%, because we now know that it is a Hereford. ‘The prior’ of 55.2% is revised to 66.7%, which is ‘the consequent’ or ‘the posterior’, based upon the further review.

Unfortunately in jargon, what is said to be determined is the degree of ‘truth’, that ‘this’ steer is branded X, prior to and posterior to its being identified as a Hereford. In jargon, the degree of ‘truth’ of the ‘belief’ that this steer is branded X is revised from 55.2% to 66.7% given the ‘truth’ that it is a Hereford.

The jargon is misleading. We are not quantifying truth. We are quantifying prejudice. Without checking to see its brand, we are prejudiced to 55.2% that this steer is branded X. Upon further review, in noting that it is a Hereford, again without checking the brand, we up our prejudice that it is branded X to 66.7%.

Outline of the Critique Presented in this Essay

Bayesian analysis starts by considering the data of both columns in identifying the ‘prior’ probability. Bayesian analysis then focuses exclusively on only one column of data in identifying the ‘posterior’ probability.

In the illustration, data of both columns, including both Hereford and Angus steers, determine the ‘prior’. The calculated ‘posterior’ applies only to the column, Herefords, H. It is the fraction of Herefords branded X.

In Lataster’s case study of Herod Agrippa’s death, Hereford, H, and Black Angus, B, are replaced with revelation, e, and background knowledge, b. The row brands, X and Y, are replaced with the brand, h, deaths by angels and the brand ~h, deaths otherwise.

In Lataster’s example, deaths by angels from both columns would include revelation, e, and background, b. to determine the ‘prior’. The calculated ‘posterior’ applies only to deaths in revelation, e.  It is the fraction of deaths in revelation, e, which are branded h, i.e. ‘deaths by angels’.

However, Lataster claims to apply Bayesian analysis by completely ignoring the column, revelation, e, rather than by focusing on it. In the illustration above, that would be like completely ignoring the column, Herefords, H, rather than by focusing on it.

Bayes’ Theorem as Applied by Lataster to Herod’s Death

Raphael Lataster employs an erroneous rationale in his illustration of the application of Bayes’ theorem to biblical history. His rationale denies his own initial positing of the constituents of a tabulated population. His rationale devolves the population into a one by two ID system, thereby rendering Bayes’ theorem inapplicable.

Lataster implicitly proposes the following table. The first column, whose ID is revelation or e, lists the numbers from the bible. The second column, whose ID is background or b, lists the numbers from common knowledge. The ID of the first row is Deaths by Angels, h. The ID of the second row is Deaths Otherwise, ~h. Lataster’s analysis is numerically less specific than this table.

Given the tabulated data, it is not necessary to apply Bayes’ theorem. Of deaths reported in revelation, the fraction claimed to be due to angels is 185,001 / (MBN  + 185,001). However, Bayes’ theorem assumes the tabulated data are not known, but some fractions are.

Lataster implicitly proposes the following rationale: From background information, the fraction of deaths, which are due to angels is virtually 0. To form the numerator of the prior fraction in Bayes’ theorem, the deaths by angels reported in revelation are added to the 0 of background deaths by angels. However, the denominator, which is total deaths from both revelation and background, is so large that the fraction, i.e. ‘the prior’ is virtually 0. Therefore, the calculated consequent or posterior fraction is virtually 0. ‘The posterior’ is the fraction of all of the deaths in revelation, which are branded h in revelation, i.e. as deaths by angels. Note: the focus of the conclusion is the column, revelation, e.

Lataster loses sight of his implicit rationale.

“In fact, I would argue that employing Bayesian reasoning without calculations is potentially more useful and reliable, given that a multitude of errors can be made when assigning quantitative values. The inherent probability of the theory (without yet considering the available evidence, such as the reference in Acts), P(h|b), is infinitely small. Conversely, P(~h|b), is very large, rendering the possibility of h, virtually 0%.”

It is only the fact that Cell(h,e) ≥ 1, that renders the probability of h virtually, rather than actually 0. Therefore, Lataster cannot treat the value of Cell(h,e) as 0. Yet, he claims,

This means that the revealed evidence, e, did not even need to be considered in order to rationally dismiss the claim (h).

Rationally dismissing the claim (h) means denying that there could be even one death in revelation, which is branded h, ‘due to an angel’. This is a flat out contradiction of the data of column, e, namely deaths reported in revelation, which includes Herod’s death. Yet, Lataster is allegedly applying Bayes’ theorem to quantify the fractions of column e.

Lataster takes Cell(h,e) = 0, algebraically. Further, if Cell(h,e) = 0, rather than ≥1, because it, ‘did not even need to be considered’, then row h has no content, thereby completely eliminating row h. Bayes’ theorem is thereby inapplicable. This is due to the fact that the ID markers of the population would be one row by two columns and not the minimum two by two required by Bayes’ theorem.

Also, if column e ‘did not even need to be considered’, then the entire data set is contained in one homogeneous cell, namely Cell(~h,b). It is the number of deaths branded ~h, excluding those reported in revelation, e.

In the course of his case study, Lataster unwittingly and unwarrantedly changed the contents of the cell of focus, Cell(h,e) to 0. Thus, he was no longer discussing the population which he originally proposed. This also devolved the population of his study from having two by two sets of ID markers to one by two or one by one. Yet Lataster concludes:

As this case study demonstrates, Bayesian reasoning is formally and mathematically valid, even if accurate calculations are not done.

I agree that precise calculations need not be done. However, in such a case study one must at least acknowledge the constituents of the tabulated population of the case study as initially implied, as well as conform to the population format required by Bayes’ theorem. In his case study, Lataster violates both of these conditions. His track record in ‘formulating probabilistic arguments’ is not good.