Thomas Bayes, an 18th century minister in England, discovered a theory of "conditional probability" in his attempt to use mathematical equations to prove or disprove the existence of God. Bayes' Theorem was published after his death by a colleague. It allows the computation of the probability of an event that we cannot prove with exact certitude from our observations of one or more related events that are provable. This theory can be applied to the question of authorship of Dreams from My Father using the observations included in Jack Cashill's well documented book Deconstructing Obama.
Along with many subjective observations from Mr. Cashill about stylistic inconsistencies between Dreams and three known written works of Obama, together with the stylistic consistencies between Dreams and William Ayers' book Fugitive Days, there is discussion by Mr. Cashill of a quantitative observation made by a Mr. Southwest of 759 similarities between Dreams and Fugitive Days. Of these 759 similarities, Mr. Cashill categorizes 180 as "striking similarities." One example set forth in his book Deconstructing Obama is a quotation from Carl Sandburg's poem "Chicago" which correctly reads "Hog Butcher for the World," and is incorrectly quoted in both Dreams and Fugitive Days as "hog butcher to the world." The similarity here is not only the exact same misquotation, but of all the poems to quote, of all the authors to quote, of all the lines to quote, why this author, why this poem, why this line?
Mr. Cashill's assessment of the number of similarities that exist between one of his books and Fugitive Days provides all the data needed to apply Bayes' Theorem to quantify both the probability that Obama is the true author of Dreams and the probability that Ayers is the true author of Dreams.
The quantitative assessments by Mr. Cashill on the number of similarities between one of his books, Sucker Punch, a memoir dealing extensively with race much like Dreams, and Fugitive Days provides the data needed to approximate the unknown probability distribution of similarities between two books written by different authors on much the same subject. Mr. Cashill found only six definite similarities, with a maximum of sixteen possible or definite similarities. Because of the multi-dimensional aspect of identifying similarities of any kind between two books, it makes sense to apply the Central Limit Theorem and assume this unknown probability distribution is of Gaussian form. Assume based on the data provided a most probable mean value of six, and a standard deviation of 10 for the distribution of similarities between two books on similar topics by different authors.
Bayes' Theorem for a problem where the probabilities of two mutually exclusive, non-overlapping events (lets call them A1 and A2) are to be computed assuming an event B is shown below (see http://stattrek.com/Lesson1/Bayes.aspx):
Event A1= Obama is the author of Dreams
Event A2= Ayers is the author of Dreams
Event B=Discovery of 180 striking similarities between Dreams and Fugitive Days
P(A1|B) is to be read as the probability of event A1 given the event B. Similarly P(B|A1) is to be read as the probability of event B given event A1. The same for P(A2|B) and P(B|A2). P(A1) and P(A2) are the a priori probabilities of events A1 and A2 when event B is not even a glint on the horizon. When Dreams was published, the assumed probability of Obama as author was 100%. After all, the publisher listed Obama as author. How could it not be? But remember, other people have published books listing themselves as the author, and much later it was discovered that someone else had written the books for them. A famous case in point involving a former President would be Profiles in Courage. It was somewhat surprising to me initially, but made sense when I had completed the analysis, that one can assume any value for P(A1), or P(A2) for that matter, and Bayes' Theorem will produce the exact same result for the conditional probability of events A1 and A2 in this case.
The conditional probabilities of event B given events A1 or A2 are computed using the probability distribution described above for the number of similarities between two books on a similar subject by different authors. The probability of event B happening if event A1 is assumed, that is if you assume that Obama actually is the author of Dreams, is small. Very small. Using my 40 year old copy of Abramowitz and Stegun, it is one divided by ten to the power of fifty using only the "striking similarities" quoted by Mr. Cashill as event B. To give you a feel of the size of ten to the power of fifty, that is comparable to an estimate of the number of atoms in the universe. If you use all 759 similarities observed by Mr. Southwest, the probability is one divided by ten to the power of approximately twelve hundred. Both are ridiculously small numbers. Both are zero to 50 significant digits or more.
The results of my analysis using Bayes' Theorem for the conditional probabilities of events A1 and A2 are as follows:
P(A1|B)= 0.00% There is zero probability that Obama is the author of Dreams
P(A2|B)=100% There is 100% probability that Ayers is the author of Dreams
An assumption implicit in the analysis so far is that William Ayers is the one true author of Fugitive Days. Mathematically the symmetry of the similarities discovered by Mr. Southwest could be applied to question the authorship of Fugitive much as it has been applied to question the authorship of Dreams. And the result of applying Bayes' Theorem would predict, if one assumes Obama wrote Dreams, then it is virtually certain that he wrote Fugitive. What the mathematics is really proving is both Dreams and Fugitive are written by the same author with a probability of 100% out to many significant figures. That author is Ayers, Obama, or some third person yet unnamed.
In the absence of additional quantitative data looking for similarities between written works by these authors, I have to leave it to the experts in writing like Mr. Cashill to determine authorship based on motive, opportunity, and demonstrated writing ability. His book Deconstructing Obama makes a strong case that Ayers had the motive, opportunity, and demonstrated writing ability to write Dreams, and that Obama's writing ability is not up to the task of writing either book.