Verbalizing Bayes’ Theorem

Bayes’ theorem is a fraction, expressed algebraically in terms of other fractions.

The theorem applies to a set of data that may be tabulated in a two by two format. The data set consists of two rows by two columns. Tabulated data, with more than two rows and/or more than two columns, may be reduced to the two by two format. All rows, but the top row may be combined to form a single, bottom row as the complement of the top row. Similarly, all columns, but the left column may be combined to form a single, right column, as the complement of the left column.

Let the rows be labeled X and non-X. Let the columns be labeled A and non-A. The table presents four quadrants of data. Let the upper left quadrant be identified as (X,A). Let the total of row X be labeled TX, the total of column A be labeled TA and the grand total of the data be labeled T.

The Algebraic Form: Fractions

Bayes’ theorem or Bayes’ equation is,

(X,A) / TA = ((TX / T) * ((X,A) / TX)) / (TA / T) Eq. 1

The validity of Bayes’ equation can easily be demonstrated in that both T and TX cancel out on the right hand side of the equation, leaving the identity, (X,A) / TA ≡ (X,A) / TA

In accord with the fact that (X,A) + (non-X,A) = TA, the denominator, TA / T, is often expressed as,

((TX / T) * ((X,A) / TX) + (((Tnon-X) / T) * ((non-X,A) / (Tnon-X)) Eq. 2

The Verbal Form: Fractions

Verbalizing Eq. 1, we have,
Cell (X,A) as a fraction of Column A equals
(Row X as a fraction of the grand total, times Cell (X,A) as a fraction of row X) divided by column A as a fraction of the grand total.

Eq. 2, the denominator, i.e. column A as a fraction of the grand total, may be expressed as,
(Row X as a fraction of the grand total, times the Cell (X,A) as a fraction of row X) plus
(Row non-X as a fraction of the grand total, times Cell (non-X,A) as a fraction of row non-X)

Replacing the Row, Column and Element Labels

On page 50 of Proving History, Richard Carrier replaces the row, column and element labels. In place of the row labels, X and non-X, he uses ‘true’ and ‘isn’t true’. In place of the column label, A, he uses ‘our’. Instead of referring to the data elements of the table as elements, Carrier refers to them as explanations. The only data in a Bayesian analysis are the elements of the table. Consequently, the only evidence considered in a Bayesian analysis is the data. In Carrier’s terminology, the only data, thus the only ‘evidence’, are the ‘explanations’.

Carrier’s Terminology for the Fractions of Bayes’ Theorem

Probability is the fraction or ratio of a subset with respect to a set. Thus, probability is a synonym for those fractions, which are the ratio of a subset to a set. Each fraction in Bayes’ theorem is a probability, the ratio of a subset to a set.

Accordingly, Carrier uses the word, probability, for the lone fraction on the left hand side of Eq. 1. However, on the right hand side of the equation, he does not use the word, probability. He uses synonyms for probability. He refers to the ratio of probability as ‘how typical’ the subset is with respect to the set. Instead of probability, he also refers to probability as ‘how expected’ the subset is with respect to the set.

Probability and improbability are complements of one, just as the paired subsets in Bayes’ theorem are complements of the set. Thus, the probability of a subset with respect to a set may be referred to as the improbability of the complementary subset. Carrier does not use the expression, improbability. Instead of referring to the improbability of the complementary subset, he refers to ‘how atypical’ is the complementary subset.

Carrier’s Verbalization of Bayes’ Theorem

Left hand side Eq. 1

Adopting Carrier’s terminology, ‘Cell (X,A) as a fraction of Column A’ would be, ‘the probability of our true explanations with respect to our total explanations’. Carrier renders it, ‘the probability our explanation is true’. It is as if probability primarily referred to just one isolated explanation rather than a subset of explanations as a fraction of a set of explanations to which the subset belongs.

The Right Hand Side, Eq. 1, the Numerator

Adopting Carrier’s terminology, the first term of the numerator, ‘Row X as a fraction of the grand total’, would be ‘how typical all true explanations are with respect to total explanations’, i.e. the fraction is TX/T. Carrier renders it ‘how typical our explanation is’. Thus, Carrier would have it to be TA/T, rather than TX/T.

In Carrier’s terminology the second term of the numerator, ‘Cell (X,A) as a fraction of row X’ would be ‘how expected are our true explanations among the set of all true explanations’. Carrier renders it ‘how expected the evidence is, if our explanation is true’. The evidence, i.e. the data, that our explanations are true, is Cell (X,A). Carrier’s rendition is thus, ‘how expected are our true explanations among the set of our true explanations’. That would be the ratio, Cell (X,A) / Cell (X,A), and not Cell (X,A) / TX.

The Right Hand Side, Eq. 1, the Denominator as Eq. 2

The first two terms of Eq. 2 are the same as the numerator of Eq. 1. Thus, there are only two more terms to be considered, namely the two terms of Eq. 2, after the ‘plus’. The first is ‘Row non-X as a fraction of the grand total’. Adopting Carrier’s terminology, this would be, ‘‘how atypical true explanations are with respect to total explanations’, i.e. the fraction is (Tnon-X)/T, which is the improbability (i.e. the atypicality) of TX/T. Carrier renders it ‘how atypical our explanation is’. Carrier would have it to be (Tnon-A)/T, which is the improbability of TA/T, rather than the improbability of TX/T.

The other term is ‘Cell (non-X,A) as a fraction of row non-X’. Adopting Carrier’s terminology, this would be, ‘how expected are our non-true explanations among the set of all non-true explanations’. Carrier renders it, ‘how expected the evidence is, if our explanation isn’t true’. The evidence, i.e. the data, that our explanations aren’t true, is Cell (non-X,A). Carrier’s rendition is thus, ‘how expected are our non-true explanations among the set of our non-true explanations’. That would be the ratio, Cell (non-X,A) / Cell (non-X,A), and not Cell (non-X,A) / Tnon-X.

Valid, but Obscurant

Each fraction in Bayes’ theorem is a fraction, which may be expressed as a probability, but also as an improbability or an atypicality. For a Bayesian tabulation of explanations, where the top row is true and the left column is our, Bayes’ theorem is the probability of true explanations among our explanations. It is also the atypicality or the improbability of non-true explanations among our explanations. However, the words, atypicality and improbability can obscure rather than elucidate the meaning of Bayes’ theorem.

Conclusion

Bayes’ theorem can be verbalized using much of Carrier’s terminology including, probability, our, explanations, true, typical, expected and atypical. However, Carrier’s actual use of his terminology does not merely obscure, but totally obliterates the algebraic and intentional meaning of Bayes’ theorem.