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Bayes Theory

REVIEW OF RELEVANT LITERATURE AND RESEARCH

I first became interested in Bayes\' Theorem after reading Blind Man\'s Bluff, Sontag (1998). The book made mention how Bayes\' Theorem was used to locate a missing thermonuclear bomb in Spain in 1966. Furthermore, it was again used by the military to locate the missing submarine USS Scorpion (Sontag, pg. 97) that had imploded when it sank several years later. I was intrigued by the nature of the theory and wanted to know more about it. When I was reading our textbook for the class, I came across Bayes\' Theorem again, and found an avenue to do more research.

There has been much study and many, many articles, papers and books devoted to Bayesian thought and statistics. My research involved literary search at the University of Memphis through Lexis-Nexis, ABI and many other electronic sources available at the University. I read many peer reviewed papers and reviewed several books about Bayed Theorem. I searched the Internet using several search engines and found much of the same literature found through the more conventional methods at the university. Additionally, as part of my research, I conducted an in depth telephone interview with the historian at the Atomic Museum in Albuquerque N.M..

I researched the development of the theorem and its criticism, and included my findings in this paper. Probably the most useful text in understanding the Theorem, and a definitive work supporting its use, is John Earman\'s work, Bayes or Bust?: A Critical Examination of Bayesian Confirmation. This book examined the relevant literature and the development of Bayesian statistics as well as defended it from its critics.

LIST OF EQUATIONS AND ILLUSTRATIONS

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Equation 1: Bayes Theorem A1

Equation 2: Bayes Theorem of Prior Probabilities A1

Equation 3: Bayes Theorem in the example of the caner test A1

Equation 4: Bayes Theorem in the example of the caner test, with

numbers applied A1

Illustration 1: Photo of B52 Bomber A2

Illustration 2: Photo of Lost bomb found off the coast of Spain A2

CHAPTER I THOMAS BAYES

Reverend Thomas Bayes was an English theologian and mathematician born in London England in 1702. His development of what is known today as Bayes\'s Theorem contributed a powerful yet controversial tool for assessing how probable a specific event or outcome will be, based on quantitative reasoning. This form of reasoning known as conditional probabilities, has been the subject of much controversy and discussion. Many debate its usefulness as a valid scientific method. However, while it does have shortcomings as pointed but by Pearson who argues that,

It does not seem reasonable upon general grounds that we should be able on so little evidence to reach so certain a conclusion….The method is much too powerful…it invests any positive conclusion, which it is employed to support, with far too high a degree of probability. Indeed, this is so foolish…that to entertain it is discreditable (1907).

Despite such criticism, it is still used today in all areas of study. Many different forms of this theory have evolved, but for the purposes of this paper, the way of looking at a problem and its solution from the Bayes point of view, can be referred to as Bayesian. "In a weak sense, any position on the foundations of probability which permits the wide or unrestricted use of Bayes\'s theorem may be described as Bayesian (Logue 1995, pg. ix)."

Thomas Bayes\' father was one of six nonconformist ministers to be ordained in England in the 17th century. After a private education near his family home in Bunhill Field, he attended the University of Endinburgh, but never finished his degree. Like his father before him, Thomas Bayes was eventually ordained a nonconformist minister. After several years of serving with his father as a Presbyterian minister, he spent most of his career as a minister in Tunbridge Wells until his death in April 1761.

In addition to his position in the community as a minister, he also had the reputation of being "…a good mathematician." (J.J. Oconnor and E.F. Robertson) In fact, he gained prominence in the field of mathematics by writing a pamphlet defending Sir Isaac Newton from critics of his work on fluxions. As result of the pamphlet, he was nominated and subsequently elected as a Fellow of the Royal Society in 1742.

The organization known as the Royal Society was a scholarly group formed