Bayes’ Theorem, Pinocle, and Raphael Lataster
This essay identifies Bayes’ theorem, illustrates it with pinocle, and critiques Raphael Lataster’s published understanding of it.
The subject of Bayes’ theorem is a set which contains two subsets that partially overlap one another. Given the ratio of each subset to the set and the ratio of the overlap to one of the subsets, Bayes’ theorem is the equation for the ratio of the overlap to the other subset.
A set of Playing Cards would be a Bayesian Set in light of the following information. The set contains two subsets, namely a subset of Clubs and a subset of Kings, which partially overlap one another. The Overlap is the subset of cards that are Clubs as well as Kings.
Given, the ratio of the subset of Clubs to the set of Playing Cards is 1/4
Given, the ratio of the subset of Kings to the set of Playing Cards is 1/6
Given, the ratio of the Overlap to the subset of Clubs is 1/6
Bayes’ theorem is:
Overlap/Kings = (Overlap/Clubs) × (Clubs/PlayingCards) × [1/(Kings/PlayingCards)] Eq. 1
Overlap/Kings = (1/6) × (1/4) × [1/(1/6)] = 1/4 Eq. 2A
Notice that the actual size of the entire set or any subset remains unknown within the context of Bayes’ theorem. Only fractions, i.e. ratios of subsets to sets is given and calculated. You may have noted that the ratios given and calculated do not conform to a deck of standard playing cards. They do conform to a standard pinocle deck.
It may be said that having been given the fraction of generic cards that are Clubs, Bayes’ theorem enables us to calculate the fraction specifically of Kings that are Clubs. However each of these two fractions is the ratio of a subset to a set, which is the definition of probability. Consequently, it would be popular to say that given the generic probability that ‘this’ playing card is a Club, Bayes’ theorem enables the calculation of the probability that ‘this’ playing card is a Club, given the additional specification that ‘this’ playing card is a King.
A Claim of Bayesian Reasoning
Raphael Lataster (p 286) has identified as Bayesian reasoning a rationale, which has nothing to do with Bayes’ theorem and which is even mathematically illogical, as will be shown, below.
He notes that the ratio of (the number of human deaths, by angels and not reported in Acts) to (the number of all human deaths, not reported in Acts) is virtually zero.
Lataster notes that consequently, the ratio of (the number of deaths by angels reported in Acts) to the number of all human deaths, not reported in Acts) must also be virtually zero. So far, so good. However, Lataster then concludes that the probability of the report of a death by an angel in Acts is false, because, by Bayesian reasoning, such deaths are zero. This falsity, known by Bayesian reasoning, is further confirmed by the fact that angels are mythical.
Lataster’s conclusion cannot proceed from Bayes’ theorem or Bayesian reasoning. What Bayes’ theorem would calculate is simply deaths by angels reported in Acts, as a fraction of the total deaths reported in Acts, without regard to the veracity of those reports. Indeed, this ratio is not necessarily even close to zero, based on the ratios which Lataster identifies as virtually zero.
Lataster’s Argument in Terms of Playing Cards
Given a Set of Playing Cards, but not the numerical ratios above, Lataster’s argument is: Given that the ratio of (Clubs,NotKings)/( Cards,NotKings) is virtually zero, we can surmise that the ratio of (Clubs that are Kings)/(Cards,NotKings) is also virtually zero, because (Clubs that are Kings) is a smaller set than (Clubs,NotKings), where the smaller set is the ‘Overlap’. However, Lataster does not stop there. He further concludes, that it would be false to claim that the ‘Overlap’ contains even one card, because that set is virtually zero. Lataster’s argument goes from what is true of a ratio, to deduce that it must be true of the numerator alone. That is illogical.
Lataster’s ‘Added Given’ to a Deck of Cards
Lataster has added another given to the standard givens of Bayes’ theorem. It is that a specific ratio is virtually zero. To attain a set which conforms to Lataster’s ‘Added Given’, all we need do is add a trillion, trillion cards NotClubsNotKings to a pinocle deck. Conforming to Lataster’s added given, the number of Clubs,NotKings would be 10 and the number of cards NotClubs,NotKings would be 30 + one trillion, trillion or 30 + 10^24. The ratio, (Clubs,NotKings)/(Cards,NotKings) = 10/(40 + 10^24), which is virtually zero. Similarly, the ratio of (Clubs that are Kings)/(Cards,NotKings) is also virtually zero. Does this mean that the Overlap is virtually zero, which is analogous to Lataster’s claim (p 286)? The Overlap is Cards that are both Clubs and Kings.
Adding one trillion, trillion cards NotClubsNotKings to a pinocle deck, we have as given: The ratio of the subset of Clubs to the set of Playing Cards is 12/(48 + 10^24); The ratio of the subset of Kings to the set of Playing Cards is 8/(48 +10^24); and The ratio of the Overlap to the subset of Clubs is 1/6. Bayes’ theorem is:
Overlap/Kings = (Overlap/Clubs) × (Clubs/PlayingCards) × [1/(Kings/PlayingCards)] Eq. 1
Overlap/Kings = (1/6) × [12/(48 + 10^24)] × [1/{8/(48 + 10^12)}] = 1/4 Eq. 2B
Assessment of the Latasterian Argument
Lataster’s argument (p 286) is a contradiction of the premise of Bayes’ theorem that the division of a set is into two subsets, which are independent of one another. This is illustrated above by two numerical examples. In both examples, the division of a deck of playing cards is into a subset of Clubs and a subset of Kings, which subsets partially overlap The first example is a pinocle deck. In the second example, a trillion, trillion cards are added to the NotClubsNotKings subset, in accord with Lataster’s added given. The added given has no effect upon Bayes’ theorem because in a set to which Bayes’ theorem is applicable, the division into subsets is a division into subsets, which are independent of one another. Lataster’s argument (p 286) is erroneous in principle by contradicting Bayes’ theorem in principle that the subsets defined are independent of one another.
Lataster’s argument has also been numerically illustrated to be erroneous in this essay: (1) using the subsets of cards of a pinocle deck (by Bayes’ theorem, Eq. 2A = 1/4) and (2) using a pinocle deck to which one trillion, trillion cards NotClubsNotKings have been added. This modification satisfies Lataster’s ‘added given’, which identifies a ratio as ‘virtually zero’. Nevertheless, by Bayes’ theorem the ratio of (Clubs that are Kings)/(Kings) doesn’t change. (Eq. 2B = 1/4).
The Latasterian conclusion, namely, that the subset (Clubs that are Kings) is zero, does not follow. In fact, that subset is a multiple of 1. It is the numerator of the ratio of (Clubs that are Kings)/(AllKings), which ratio, by Bayes’ theorem, is 1/4 (Eq. 1B) and 1/4 (Eq. 2B).
It is true of the pinocle deck, to which a trillion, trillion NotClubs,NotKings are added, that the ratio of (Clubs,NotKings)/(CardsNotKings) is virtually zero, namely, 10/(40+10^24). It is also true that the ratio of (Clubs that are Kings)/(CardsNotKings), is virtually zero, namely, 2/(40+10^24).
Thus the two premises of Lataster are true of the set to which a trillion, trillion cards NotClubsNotKings are added. It is not that his premises are false. It is that his conclusion does not follow from his premises. It is that both of his premises are irrelevant to Bayes’ theorem. The subset, NotClubsNotKings, is not a factor of Bayes’ theorem Eq. 1, so its size, relatively small or relatively large, appears to be irrelevant. Further, the only term in Bayes’ theorem Eq. 1, affected by the size of the subset, NotClubsNotKings, is the total set of Playing Cards. However, the entire set, Playing Cards, is both in the numerator and the denominator of Bayes’ theorem Eq. 1 thereby cancelling each other out and rendering Overlap/Kings, independent of the size of Playing Cards and the size of NotClubsNotKings (Eq. 1).
If to one trillion, trillion NotClubsNotKings, we add one pinocle deck, the subset of (Clubs that are Kings) is actually 2, while the Bayes’ theorem ratio of Overlap/Kings is 1/4. If we were to add three pinocle decks to the one trillion, trillion NotClubsNotKings, the subset of (Clubs that are Kings) would be 6, while the Bayes’ theorem ratio of Overlap/Kings would still be 1/4. This illustrates the independence of the subset of Kings and the subset of NotKings. This independence is true of Bayesian sets. It is this independence that is denied by Lataster’s argument (p 286). He assumes that a characteristic within the subset, NotKings, must be characteristic of the subset, Kings.
Conclusion
Latasterian reasoning is illogical. The fact that a ratio is virtually zero, does not mean that the numerator of the ratio is actually zero, which is what Lataster (p 286) illogically concludes. In the illustration of this essay, the ratio of (Clubs that are Kings)/(CardsNotKings) is virtually zero, namely, 2/(40 + 10^24). It would be illogical to conclude that the numerator must be actually zero. In the illustration the numerator is 2.
The subject of Latasterian reasoning are two ratios, neither of which is any of the four ratios of Bayes’ theorem Eq. 1. Thus, Latasterian reasoning is not Bayesian reasoning. It is irrelevant to Bayesian reasoning. In the illustration, the two ratios, which Lataster subjects to his mode of reasoning are: (ClubsNotKings)/(CardsNotClubs) and (Clubs that are Kings)/( CardsNotClubs). Neither of these is any of the four ratios of Bayes’ theorem, Eq. 1. These two Latasterian ratios are irrelevant to Bayes’ theorem and Bayesian reasoning.
Each of the four ratios of Bayes’ theorem is a probability, i.e. it is a ratio of a subset to a set (Eq. 1). In contrast, only one of the two ratios of Latasterian reasoning is a probability. Of the two ratios subjected to Latasterian reasoning, only (ClubsNotKings)/(CardsNotKings) is a probability, i.e. the ratio of a subset to a set. The other ratio is (Clubs that are Kings)/(CardsNotClubs). The numerator is not a subset of the denominator within the context of the entire set of cards. Consequently, the ratio is not a probability.
It would be logical to identify angels and their activity as mythological, i.e. not true, and conclude thereby that a report of a death by an angel in Acts is false. In contrast, it is not logically possible to reach that conclusion based on Bayes’ theorem or Bayesian reasoning, where the entire set is human deaths, one subset is deaths by angels, and the other subset is deaths reported in Acts. The Overlap is deaths by angels, reported in Acts.
As a religious studies professional, Raphael Lataster is keenly interested in the rationale of probabilities (minute 30:45). His interest should lead eventually to his understanding of Bayes’ theorem, which is an equation of four ratios, each of which is a probability. A good starting point toward his or anyone’s understanding of Bayes’ theorem would be to compare it to Pythagoras’ theorem of which we are all familiar. I wish him well in his pursuit of understanding Bayes’ theorem.
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