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Since World War 2, a body of knowledge had developed known as Bayesian decision theory whose purpose is to provide a solution of problems involving decision making under uncertainty. Now a days Bayesian probability is known as “Bayes theorem” named after a British Mathematician Thomas Bayes (1702-1761) and published in 1763 has become one of the most memoirs in the history of mathematics and being under various controversies. The contribution of this theory consists primarily of a unique method of calculating conditional probabilities. This Bayes theorem approach used to solve problems of determining the probability of some events,A,given that another event B ,has been observed i.e. determining the value of P(A/B).The event A is usually thought of as sample information so that Bayesian’s rule is concerned with determining the probability of an event given certain sample information. For ex.A sample output of 2 defectives in 50 trials(event A) might be used to estimate that machine is not working properly (event B) or you may use the results of your first examination in statistics (event A) as sample evidence in estimating the the probability of getting first class(event B).
Bayesian Probability is based on formula for conditional probability. Let’s see:
A1 and A2 =A set of events that are mutually exclusive (the two events cannot occur together) and exhaustive (the combination of two events in entire experiment), and
B = A sample event which intersects each of events, as shown in the diagram below:
Observe the diagram. The part B which is within A1 represents the area “A1 and B” and A2 represents “A2 and B”
Then the probability of an event A1 given event B, is
P (A1/B) = P (A1 and B)/P (B)
And, similarly the probability of an event A2 givenevent B,is
P (A2/B) =P (A2 and B)/P (B)
Where , P (B) = P (A1 and B) + P (A2 and B),
P (A1 and B) = P (A1) * P (B/A1), and
P (A2 and B) = P (A2)* P (A2/B)
Probabilities that are being revised before by Bayes’ rule arecalled simply prior probabilities or priori because they are determined before these sample information is taken into consideration. A probability which has gone under revision of Bayes’rule is called posterior probability; it is also called revised probabilities because they are getting by revising the prior probabilities by gaining additional information. Revisedprobabilities are always conditional probabilities.
Some interesting points worth noting about Bayesian’ probability:
It deals with conditional probability; its interpretation is different from general conditional probabilities .The general conditional probabilities ask “what is the probability of experimental results and sample given the state value? Where Bayesian probabilities asks “what is the probability of state value given the experimental result and sample?
When we are talking about Bayesian probability, different decision- maker may assign different probabilities to same set of states of nature. We may also conduct experiment by using the posterior probabilities of previous experiment as prior probabilities. The more evidence we accumulate, prior probabilities became less important.
The notion of prior and posterior in Bayesian’ probability is relative to given sample outcome.
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