Bayes Solution Bayes\' theorem deals with the role of new information in revising probability estimates. The theorem assumes that the probability of a hypothesis (the posterior probability) is a function of new evidence (the likelihood) and previous knowledge (prior probability). The theorem is named after Thomas Bayes (1702–1761), a nonconformist minister who had an interest in mathematics. The basis of the theorem is contained in as essay published in the Philosophical Transactions of the Royal Society of London in 1763. Bayes\' theorem is a logical consequence of the product rule of probability, which is the probability (P) of two events (A and B) happening— P(A,B)—is equal to the conditional probability of one event occurring given that the other has already occurred—P(A|B)—multiplied by the probability of the other event happening—P(B). The derivation of the theorem is as follows: P(A,B) = P(A|B)×P(B) = P(B|A)×P(A) Thus: P(A|B) = P(B|A)×P(A)/P(B). Bayes\' theorem has been frequently used in the areas of diagnostic testing and in the determination of genetic predisposition. For example, if one wants to know the probability that a person with a particular genetic profile (B) will develop a particular tumour type (A)—that is, P(A|B). Previous knowledge leads to the assumption that the probability that any individual will develop the specific tumour (P(A)) is 0.1 and the probability that an individual has the particular genetic profile (P(B)) is 0.2. New evidence establishes that the probability that an individual with the tumor—P(B|A)—has the genetic profile of interest is 0.5. Thus: P(A|B) = 0.1×0.5/0.2 = 0.25 The adoption of Bayes\' theorem has led to the development of Bayesian methods for data analysis. Bayesian methods have been defined as \"the explicit use of external evidence in the design, monitoring, analysis, interpretation and reporting\" of studies (Spiegelhalter, 1999). The Bayesian approach to data analysis allows consideration of all possible sources of evidence in the determination of the posterior probability of an event. It is argued that this approach has more relevance to decision making than classical statistical inference, as it focuses on the transformation from initial knowledge to final opinion rather than on providing the \"correct\" inference. In addition to its practical use in probability analysis, Bayes\' theorem can be used as a normative model to assess how well people use empirical information to update the probability that a hypothesis is true. Oxford Dictionary of Philosophy: Theorem in probability theory. Thomas Bayes (1702-61) was an English clergyman, whose An Essay towards Solving a Problem in the Doctrine of Chances occurs in two memoirs presented by Price (Bayes having died), in Philosophical Transactions of 1763 and 1764. Bayes gave a result for the probability that the chance of an event on a single trial is within a certain interval, given the number of times the event has occurred and the number.

1. Bayes
Solution
Bayes' theorem deals with the role of new information in revising probability
estimates. The theorem assumes that the probability of a hypothesis (the posterior probability) is
a function of new evidence (the likelihood) and previous knowledge (prior probability). The
theorem is named after Thomas Bayes (1702–1761), a nonconformist minister who had an
interest in mathematics. The basis of the theorem is contained in as essay published in the
Philosophical Transactions of the Royal Society of London in 1763. Bayes' theorem is a logical
consequence of the product rule of probability, which is the probability (P) of two events (A and
B) happening— P(A,B)—is equal to the conditional probability of one event occurring given that
the other has already occurred—P(A|B)—multiplied by the probability of the other event
happening—P(B). The derivation of the theorem is as follows: P(A,B) = P(A|B)×P(B) =
P(B|A)×P(A) Thus: P(A|B) = P(B|A)×P(A)/P(B). Bayes' theorem has been frequently used in
the areas of diagnostic testing and in the determination of genetic predisposition. For example, if
one wants to know the probability that a person with a particular genetic profile (B) will develop
a particular tumour type (A)—that is, P(A|B). Previous knowledge leads to the assumption that
the probability that any individual will develop the specific tumour (P(A)) is 0.1 and the
probability that an individual has the particular genetic profile (P(B)) is 0.2. New evidence
establishes that the probability that an individual with the tumor—P(B|A)—has the genetic
profile of interest is 0.5. Thus: P(A|B) = 0.1×0.5/0.2 = 0.25 The adoption of Bayes' theorem
has led to the development of Bayesian methods for data analysis. Bayesian methods have been
defined as "the explicit use of external evidence in the design, monitoring, analysis,
interpretation and reporting" of studies (Spiegelhalter, 1999). The Bayesian approach to data
analysis allows consideration of all possible sources of evidence in the determination of the
posterior probability of an event. It is argued that this approach has more relevance to decision
making than classical statistical inference, as it focuses on the transformation from initial
knowledge to final opinion rather than on providing the "correct" inference. In addition to its
practical use in probability analysis, Bayes' theorem can be used as a normative model to assess
how well people use empirical information to update the probability that a hypothesis is true.
Oxford Dictionary of Philosophy: Theorem in probability theory. Thomas Bayes (1702-61) was
an English clergyman, whose An Essay towards Solving a Problem in the Doctrine of Chances
occurs in two memoirs presented by Price (Bayes having died), in Philosophical Transactions of
1763 and 1764. Bayes gave a result for the probability that the chance of an event on a single
trial is within a certain interval, given the number of times the event has occurred and the number

2. it has failed. But the form in which his theorem is remembered is as an expression for the
posterior probability of a hypothesis (its probability after evidence is obtained). This is a product
of (i) its probability before the evidence, or prior probability, and (ii) the probability of the
evidence being as it is, given the hypothesis, divided by (iii) the prior probability of the evidence
(often expressed as the probability of the evidence considered in the light of all the different
possible hypotheses).