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Bayes' Theorem Definition
Bayes' theorem which is also known as Bayes' law is a theorem with two distinct interpretations. In the Bayesian interpretation, it shows how a subjective degree of belief should rationally change to account for evidence and in the frequentist interpretation, it is used to relate inverse representations of the probabilities concerning two events. In the Bayesian interpretation, the basis of Bayesian statistics is Bayes' theorem which has applications in almost all the fields. This theorem is now widely accepted and this may have been due to the development of computing which enabled the successful application of Bayesianism to many complex problems.
For events and, provided that
In a Bayesian inference, the probability of evidence is constant for all models and the posterior may then be expressed as proportionality to the numerator as follows:
In the extended form, for some partition of a set of the event space where the event space is given or conceptualized in terms of and It is then used to eliminate using the Law of total probability which is represented as follows:
In the special case of a binary partition it is represented as follows:
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Random variables
Consider a sample space which is generated by two random variables that are and. In principle, Bayes' theorem applies to the events and then the terms become 0 at points where either variable has finite probability density function. To remain useful, Bayes' theorem may be formulated in terms of the relevant densities.
In its simple form, is continuous and is discrete and it takes the following form:
It is discrete and is continuous it is represented as follows:
If both and are continuous then it is represented as follows:
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In the extended form, a continuous event space is often conceptualized in terms of the numerator terms and so it is then used to eliminate the denominator using the Law of total probability. For, this becomes an integral as follows:
Bayes' rule states that under the Bayesian probability, the Bayes' rule may be thought of as Bayes' theorem in odds and it is represented as follows:
where
So the rule states that the posterior odds are the prior odds times the Bayes factor.
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