Bayes and Blueberries; or, Say “No” to Algebra

“Nix algebra, deaths, and vampires” was the message I received in comments on my previous essay. I am omitting the algebra from this essay. In this essay, I am also saying “No” to deaths and vampires. To replace them, I am saying “Yes” to cereal and cereal topped with blueberries. Without algebraic symbols, this essay will demonstrate the illogic of Raphael Lataster’s “Bayesian reasoning” (p. 286) by which he attempted to debunk biblical literature, namely the report of a death by an angel in Acts 12:21-23.

One Day at Morning Break

During a lull in the chatter at a morning break, Bill Bayes said he had come across a mathematical relationship among sets and subsets called Bayes’ theorem. It’s subject is any set with two subsets, which partially overlap one another. Given the numerical values of the ratio of each of the subsets to the set and the numerical value of the ratio of the overlap to one subset, one can calculate the numerical value of the ratio of the overlap to the other subset using Bayes’ theorem.

Take all the bowls of cereal served in restaurants in the U.S. in 2019 as the set. If given the fraction of all bowls that were topped with blueberries, the fraction of all bowls that were hot cereal, and the fraction of those with blueberries that were hot cereal, one can use Bayes’ theorem to calculate the fraction of bowls of hot cereal that were topped with blueberries.

The Next Day at Work Break

Chuck Itout said he was fascinated with Bill’s comments and surfed the internet for hot cereal and blueberries. He came across the claim that the Association of American Restauranteurs (AAR) served hundreds of bowls of hot cereal with blueberries in 2019. Chuck noted that wasn’t enough information to employ the equation that is Bayes’ theorem. But it was enough, along with common knowledge, to apply Bayesian reasoning and thereby demonstrate that the AAR’s claim of serving hundreds of bowls of hot cereal with blueberries in 2019 was false.

Chuck noted that everyone knows that almost all cereal served in the US in restaurants is cold cereal without blueberries. The subset of cold cereal is not only virtually without blueberries, but it dwarfs the hot cereal subset. Because the cold cereal fraction so overwhelms the hot cereal fraction we can employ Bayesian reasoning to deduce that the number of bowls of hot cereal served with blueberries in restaurants in 2019 has got to be virtually zero. Thus, the AAR’s claim is false.

Mary was quite contrary. She told Chuck that his rationale had nothing to do with Bayes’ theorem or Bayesian reasoning. But, foremost, it was mathematically illogical. She noted that what is true of an entire set is true of every subset. In contrast, what is true of one subset, may or may not be true of another subset. It doesn’t matter how overwhelmingly large the first subset is relative to the other subset.

Suppose every bowl of hot cereal served in a restaurant in 2019 had blueberries on top. Such would be compatible with the fact that no one has ever been served cold cereal with blueberries. It would also be compatible with the fact that the ratio of bowls of hot cereal to bowls of cold cereal was virtually zero. Bayes’ theorem calculates a ratio within the subset of hot cereal, which is completely independent of any ratio within cold cereal, as well as being independent of the ratio of hot cereal to cold cereal.

Conclusion

Chuck’s claim of employing Bayesian reasoning mathematically to debunk the report of the AAR of serving hundreds of bowls of hot cereal with blueberries in 2019 is illogical. In parallel, Raphael Lataster’s claim of employing Bayesian reasoning mathematically to debunk a report of a death by an angel in Acts is illogical. That is not to say that either report is therefore true.

The mathematics of sets is not very complicated. However, algebraic symbols may give the appearance of complexity. Consequently, they were not used in this essay. Death, whether or not by an angel, is a dour topic. Breakfast cereal with or without blueberries is a pleasant topic. Cereal is the subject of the analysis of sets and subsets in this essay to help focus on the logical error. It is the same logical error of Raphael Lataster in his mathematical argument, allegedly based on Bayesian reasoning, in support of probabilism (a form of relativism) as characteristic of human knowledge.

Parallels: “with blueberries” and “by angels”

Here is a list of the parallel IDs of the set and subsets of Bayes’ theorem in this essay (TE) and of the “Bayesian reasoning” of Lataster (L) in his essay (p. 286):

• TE’s Main Set: all bowls of cereal (served in US restaurants in 2019)
  • L’s Main Set: all human deaths
• TE’s First Subset: all bowls of cereal with blueberries
  • L’s First Subset: (h) all deaths by angels
• TE’s Second Subset: all bowls of hot cereal
  • L’s Second Subset: (e) all deaths reported in Acts
• TE’s Over-lapping Subset: bowls of cereal with blueberries which are also hot cereal
  (The overlap belongs to both subsets: all cereal with blueberries and all hot cereal)
  • L’s Over-lapping Subset: (e,h) deaths reported in Acts as deaths by angels
    (The overlap belongs to both subsets: (h) all deaths by angels and (e) all deaths reported in Acts)

That which can be calculated by Bayes’ theorem is the overlap as a fraction of the second subset.

• TE’s: That which can be calculated by Bayes’ theorem: the overlap, bowls of hot cereal “with blueberries”, as a fraction of the subset, all bowls of hot cereal
• L’s: That which can be calculated by Bayes’ theorem: the overlap, (e,h) deaths “by angels” reported in Acts, as a fraction of subset, (e), all deaths reported in Acts

Bayes’ theorem enables the calculation of a ratio based on given data. It cannot critique the data given as the fictional Chuck Itout attempted in this essay and as religious studies professional Raphael Lataster, attempted in his essay (p. 286).