Реферат: Bayes Theory Essay Research Paper REVIEW OF

Bayes Theory Essay, Research Paper

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

page

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 to promote the natural sciences, including mathematics and all applied aspects such as engineering, and medicine. The society was founded in 1660 during the reign of King Charles II, and was incorporated by royal charter in 1662. The society is self-governed by a president and council, whose statutory responsibilities include making appointments to research councils, and it has representatives in the governing bodies of many organizations. The people that nominated Bayes described him as, “a gentleman of known merit, well skilled in Geometry and all parts of Mathematical and philosophical Learning (Norland, 2000 pg. 2)”. Bayes retired from the ministry in 1752 and died nine years later.

CHAPTER II DEVELOPMENT OF BAYES THEOREM

After Bayes death in 1761, his family was left with most of his property, but he also left a small bequest to Richard Price, another minister and amateur mathematician. Among Bayes’ papers, Price found two essays on mathematical subjects. He was so impressed with them that he sent them to the Royal Society hoping they would be published. Bayes set out his theory of probability in one of the essay’s titled towards solving a problem in the doctrine of chances it was so well received by the Royal Society, that it was published in the Philosophical Transactions of the Royal Society of London in 1764.

Bayes essay would shape the nature of statistics. Bayes’ theory stated “Given the number of times in which an unknown event has happened and failed: Required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named (Bayes 1763, pg. 376)”. This problem and the solution it entails have come to be referred to as inverse statistics.

Logue put it this was,

Thus Bayes, in his famous [1763] essay, having defined probability as ‘the ratio between the value at which the expectation depending upon the happening of the event ought to be computed, and the value of the thing expected upon its happening’, then equivocates upon ‘expect’, sometimes taking it as wholly relative to an individual’s mental state, sometimes as though expectations were externally fixed values. (1995, pg.95)

The theorem was eventually accepted by mathematicians of the time. The mathematician LaPlace later accepted it as a valid process as Jaynes (1995) points out, “In almost his first published work (1774), Laplace rediscovered Bayes’ principle in greater clarity and generality, and then for the next 40 years proceeded to apply it to problems of astronomy, geodesy, meteorology, populations statistics and even jurisprudence” (pg. 2). LaPlace generalized Bayes’ approach, which was later generalized further into what we now call Bayes’ theorem. Essentially, the theorem is supposed to quantify the value of a hunch, factor in the knowledge that exists in people beyond their conscious minds. You see, according to Bayes’ theorem, one can always start with a belief with regard to the probability of an outcome and use that in the equation. If one has no prior knowledge, the prior distribution would be diffuse (spread out).

CHAPTER III BAYES THEOREM EXPLAINED

Bayes theorem for conditional probabilities is described by equation 1:

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