[名人] Thomas Bayes
※ [本文轉錄自 Math 看板]
作者: fizeau (.) 看板: Math
標題: [名人] Thomas Bayes
時間: Sat May 3 12:47:10 2008
http://www.mrs.umn.edu/~sungurea/introstat/history/w98/Bayes.html
English theologian and mathematician Thomas Bayes has greatly contributed to
the field of probability and statistics. His ideas have created much
controversy and debate among statisticians over the years.
Thomas Bayes was born in 1702 in London, England. There appears to be no
exact records of his birth date. Bayes's father was one of the first six
Nonconformist ministers to be ordained in England. (4) Bayes's parents had
their son privately educated. There is no information about the tutors Bayes
worked with. However, there has been speculation that he was taught by de
Moivre, who was doing private tuition in London during this time.
Bayes went on to be ordained, like his father, a Nonconformist minister. He
first assisted his father in Holborn, England. In the late 1720's, Bayes took
the position of minister at the Presbyterian Chapel in Tunbridge Wells, which
is 35 miles southeast of London. Bayes continued his work as a minister up
until 1752. He retired at this time, but continued to live in Tunbridge Wells
until his death on April 17, 1761. His tomb is located in Bunhill Fields
Cemetery in London.
Throughout his life, Bayes was also very interested in the field of
mathematics, more specifically, the area of probability and statistics. Bayes
is believed to be the first to use probability inductively. He also
established a mathematical basis for probability inference. Probability
inference is the means of calculating, from the frequency with which an event
has occurred in prior trials, the probability that this event will occur in
the future. (5) According to this Bayesian view, all quantities are one of
two kinds: known and unknown to the person making the inference. (6) Known
quantities are obviously defined by their known values. Unknown quantities
are described by a joint probability distribution. Bayesian inference is seen
not as a branch of statistics, but instead as a new way of looking at the
complete view of statistics. (6)
Bayes wrote a number of papers that discussed his work. However, the only
ones known to have been published while he was still living are: Divine
Providence and Government Is the Happiness of His Creatures (1731) and An
Introduction to the Doctrine of Fluxions, and a Defense of the Analyst
(1736). The latter paper is an attack on Bishop Berkeley for his attack on
the logical foundations of Newton's Calculus. Even though Bayes was not
highly recognized for his mathematical work during his life, he was elected a
Fellow of the Royal Society in 1742.
Perhaps Bayes's most well known paper is his Essay Towards Solving a Problem
in the Doctrine of Chances. This paper was published in the Philosophical
Transactions of the Royal Society of London in 1764. This paper described
Bayes's statistical technique known as Bayesian estimation. This technique
based the probability of an event that has to happen in a given circumstance
on a prior estimate of its probability under these circumstances. This paper
was sent to the Royal Society by Bayes's friend Richard Price. Price had
found it among Bayes's papers after he died. Bayes's findings were accepted
by Laplace in a 1781 memoir. They were later rediscovered by Condorcet, and
remained unchallenged. Debate did not arise until Boole discovered Bayes's
work. In his composition the Laws of Thought, Boole questioned the Bayesian
techniques.
Boole's questions began a controversy over Bayes's conclusions that still
continues today. In the 19th century, Laplace, Gauss, and others took a great
deal of interest in this debate. However, in the early 20th century, this
work was ignored or opposed by most statisticians. Outside the area of
statistics, Bayes continued to have support from certain prominent figures.
Both Harold Jeffreys, a physicist, and Arthur Bowley, an econometrician,
continued to argue on behalf of Bayesian ideas. (1) The efforts of these men
received help from the field of statistics beginning around 1950. Many
statistical researchers, such as L. J. Savage, Buno do Finetti, Dennis
Lindley, and Jack Kiefer, began advocating Bayesian methods as a solution for
specific deficiencies in the standard system. (1)
However, some researchers still argue that concentrating on inference for
model parameters is misguided and uses unobservable, theoretical quantities.
(1) Due to this skepticism, some are reluctant to fully support the Bayesian
approach and philosophy.
A specific contribution Thomas Bayes made to the fields of probability and
statistics is known as Bayes Theorem. It was first published in 1763, two
years after his death. It states:
P(H/E, C) = P(H/C) P(E/H, C) / P(E/C)
It uses probability theory as logic and serves as a starting point for
inference problems. (3) It is still unclear what Bayes intended to do with
this calculation.
The left hand side of the equation is known as the posterior probability. It
represents the probability of a hypothesis H when given the effect of E in
the context of C. The term P(H/C) is called the prior probability of H given
the context of C by itself. The term P(E/H, C) is known as the likelihood.
The likelihood is the probability of E assuming that H and C are true.
Lastly, the term 1 / P(E/C) is independent of H and can be seen as a scaling
constant. (3)
Bayes Theorem can be derived from the Product Rule of probability. The
Product Rule is P(A, B/I) = P(A/B, I) * P (B/I) = P(B/A,I) * P(A/I).
Rearranging this and extending the rule to multiple sequential updates gives:
(3)
P(H/E1,E2,E3,C) = P(H/C)*P(E1,E2,E3/H,C) / P(E1,E2,E3/C)
P(h/C)*P(E1/H,C) * P(E2/E1,H,C) * P(E3/E2,E1,H,C)
= --------------------------------------------------
P(E1/C) * P(E2/E1,C) * P(E3/E2,E1,C)
This becomes very complicated because as each new piece of E is brought into
the equation, the effect is conditional on all previous E. However, making
the assumption P(E2/E1,C) = P(E2/C) and P(E1/E2,C) = P(E1/C) avoids this
difficulty. This assumes given C knowing E2 gives no information about E1 and
vice versa. (3) The product rule then reduces to P(E1,E2/C) = P(E1/C) *
P(E2/C). (3) This must be used carefully though. Conditional Independence
does not always hold. These principles are what much of the controversy is
centered around.
As you can see, Thomas Bayes has made many important contributions to the
development of probability and statistics. Although his work has been
controversial, it has brought forth many new ideas that the world of
mathematics continues to research and benefit from.
References
1. Bradley, P. & Louis, T. (1996). Bayes and Empirical Bayes Methods for
Data Analysis. London: Chapman & Hall.
2. http://ic.arc.nasa.gov/ic/projects/bayes-group/html/bayes-theorem.html
3. http://ic.arc.nasa.gov/ic/projects/bayes-group/html/bayes-theorem-long.html
4. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bayes.html
5. http://www.stat.ucl.ac.be/ISpersonnel/beck/bayes.html
6. Johnson, N. & Kotz, S. (1982). Encyclopedia of Statistical Sciences,
1, 197-205. New York: John Wiley & Sons, Inc.
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