The simplest population or set of elements, to which Bayes’ Theorem applies, is one in which the set is divided into two subsets by each of two independent criteria or markers. One marker may be viewed as dividing the set into two rows, while the other marker divides the set into two columns. This results in the set of elements being divided into four quadrants or cells, where each cell is identified by a row marker and a column marker. Each cell is a subset of the total set.

In addition to the four subsets of the total set, which subsets are identified by row and column, another four subsets of the total set are identifiable. Two of these subsets are the sum of each row. The other two subsets are the sum of each column.

In Bayes’ theorem, the row marker is viewed as identifying one row as the Row, and the other row as non-Row. Similarly, the column marker is viewed as identifying one column as the Column, and the other column as non-Column. The specificity of each cell is by its row and column. One specific example from the following table is the cell, Cell(Row, non-Column) or Cell(true, non-our).

Probability is the ratio of a subset to a set. Consequently, due to the Bayesian divisions of the set of elements, sixteen probabilities may be identified:

Four probabilities are the ratios of each of the four cells as a subset to the total set.

Four probabilities are the ratios of each of the four cells as a subset to its row sum as a set.

Four probabilities are the ratios of each of the four cells as a subset to its column sum as a set.

Two probabilities are the ratios of each row sum as a subset to the total set.

Two probabilities are the ratios of each column sum as a subset to the total set.

Bayes’ theorem concerns only four of these ratios or probabilities. Bayes’ theorem is an equation for one of these probabilities as equaling an algebraic expression involving three of the other probabilities.

The subset for which Bayes’ theorem is the focus is Cell(Row, Column). The set of focus is Column Sum. The probability of focus is their ratio of subset to set, namely

Cell(Row, Column) / Column Sum.

Let us call this Probability One. Bayes’ theorem permits the numerical calculation of Probability One, if the numerical values of three other probabilities are known. The other three probabilities are:

Probability Two: Row Sum / Total

Probability Three: Cell(Row, Column) / Row Sum

Probability Four: Column Sum / Total

Bayes’ equation is:

Probability One = (Probability Two * Probability Three) / Probability Four

On page 50 of *Proving History*, Richard Carrier considers a Bayesian population of explanations. The Row is true. The non-Row is non-true. The Column is our. The non-Column is non-our. The four probabilities of Bayes theorem in terms of this particular Bayesian population of elements are:

One: The probability of a true explanation of ours with respect to our total explanations

Two: The probability of a true explanation with respect to total explanations

Three: The probability of a true explanation of ours with respect to total true explanations

Four: The probability of an explanation of ours with respect to total explanations.

Four is, of course, Column Sum / Total. However,

Column Sum = Cell(Row, Column) + Cell(non-Row, Column)

Therefore, Probability Four may expressed as:

Cell(Row, Column) / T + Cell(non-Row, Column) / T Eq. 1

We can multiply and divide the first ratio of Eq. 1 by Row Sum without changing the ratio. This yields:

(Row Sum / Total) * (Cell(Row, Column) / Row Sum)

This is Probability Two times Probability Three.

We can multiply and divide the second ratio of Eq. 1 by non-Row Sum without changing the ratio. This yields:

(non-Row Sum / Total) * (Cell(non-Row, Column) / non-Row Sum).

We now have all of the terminology to verbalize Bayes’ theorem in the manner of Carrier, in his example of page 50 of *Proving History*.

A lucid verbalization of Bayes’ Theorem for the set of explanations defined by Carrier is:

In contrast, Carrier’s verbalization of Bayes’ Theorem from page 50 of Proving History is obscure:

Probability is the ratio of a subset to a set. When a set is composed of discrete elements such as explanations, the probability is the ratio of the number of elements in the subset to the number of elements in the set.

In the lucid version of Bayes’ theorem, each of the five verbal probabilities is of the form: ‘the probability of an element of a subset with respect to the elements of a set’. This has the same meaning as the form ‘The ratio of the elements of a subset to the elements of a set’. For each of the five probabilities, the subset numerator and the set denominator of the ratio are clearly identified in the lucid verbalization of Bayes’ theorem.

Carrier’s verbalization uses the words, typical and expected, as synonyms for probability and the word, atypical, as a synonym for improbability. Also, the only evidence is the data, the elements of the Bayesian data set. Carrier has identified the elements of the set as explanations. Therefore, Carrier’s verbalization uses the words, evidence and explanations, as synonyms.

In Carrier’s verbalization of Bayes’ theorem, if probability, typical, expected and atypical are replaced with ratio, it is difficult, but not impossible to determine the numerator and the denominator of each ratio specified. However, these ratios are not valid expressions of the probabilities of Bayes’ theorem. Thus, Carrier’s verbalization is invalid.

Consider Carrier’s verbal expressions of the five ratios, each purportedly a probability specified by Bayes’ theorem.

First: “The probability our explanation is true”.

This literally designates the ratio of our total explanations to total true explanations. However, neither our total explanations nor total true explanations is a subset of the other. The ratio verbalized is not a probability.

Also, “The probability our explanation is true” cannot refer to or be a test of a particular explanation of ours, because probability refers to sets of elements, not to a particular element. If the numerical value of Bayes’ equation were 1, then each particular explanation of ours would be true, because all are true. However, this is an inference from a calculated result, not the implication of Bayes’ theorem in its algebraic expression, which accommodates any numerical value from 0 to 1.

Second: “How typical our explanation is”.

This literally designates the ratio of our total explanations to total explanations. Our total explanations is a subset of total explanations. That ratio is Probability Four above. However, Probability Four is the denominator of Bayes’ theorem, not an expression in its numerator as it is in Carrier’s verbalization.

Third: “How expected the evidence is, if our explanation is true”.

As noted, the only evidence is the elements of the data set. The first phrase of this quotation, “How expected the evidence is”, would then read, ‘The probability of an element’. Every probability is the probability of an element of a subset. Thus, “How expected is the evidence” can be replaced with ‘the probability’. We then have ‘the probability, if our explanation is true’. This is of the form, ‘the probability if A is B’. ‘A is B’ means A is a subset of B, which alludes to the probability of A with respect to B. The entire expression, “How expected the evidence is, if our explanation is true”, is, thus, reduced to ‘the probability our explanation is true’, which is identical to the expression of the First consideration. The ratio verbalized is not a probability.

Fourth: “How atypical our explanation is”.

This literally designates the ratio of non-our total explanations to total explanations. This ratio verbalized is a probability.

Fifth: “How expected the evidence is, if our explanation isn’t true”.

In line with the rationale above in the Third and First considerations, this can be reduced to ‘the probability of our total explanations with respect to total non-true explanations’. However, neither our total explanations nor total non-true explanations is a subset of the other. The ratio verbalized is not a probability.

Another thing to note is that in the lucid verbalization, only the second of the probabilities in the numerator of Bayes’ theorem identifies a subset as ‘our’. Yet, in Carrier’s verbalization, each of the two expressions for probability (one referred to as a typicality, the other as an expectation) in the numerator of Bayes’ theorem identify a subset as ‘our’.

There is a marked contrast between the clarity of specification of the five ratios of probability in the lucid verbalization of Bayes’ theorem and the obscurity of the ratios in Carrier’s verbalization. Three out of Carrier’s five verbal ratios are not probabilities.

Bayes’ theorem is not the equation for ‘The probability our explanation is true’, in Carrier’s words. Rather, Bayes’ theorem is the equation for ‘Among our explanations, the probability that an explanation is true’. It is the ratio of our true explanations to our total explanations.

Just as I have previously concluded from a different perspective of his verbalization: “Carrier’s actual use of his terminology does not merely obscure, but totally obliterates the algebraic and intentional meaning of Bayes’ theorem.”