Statistics Probability Bayes Theorem – lesscss

By | October 20, 2019

Statistics – Probability Bayes Theorem

One of the most significant developments in the probability field
has been the development of Bayesian decision theory which has
proved to be of immense help in making decisions under uncertain
conditions. The Bayes Theorem was developed by a British
Mathematician Rev. Thomas Bayes. The probability given under
Bayes theorem is also known by the name of inverse probability,
posterior probability or revised probability. This theorem finds
the probability of an event by considering the given sample
information; hence the name posterior probability. The bayes
theorem is based on the formula of conditional probability.

conditional probability of event ${A_1}$ given event ${B}$ is

${P(A_1/B) = frac{P(A_1 and B)}{P(B)}}$

Similarly probability of event ${A_1}$ given event ${B}$ is

${P(A_2/B) = frac{P(A_2 and B)}{P(B)}}$

Where

${P(B) = P(A_1 and B) + P(A_2 and B) \[7pt] P(B) = P(A_1)
times P (B/A_1) + P (A_2) times P (BA_2) }$

${P(A_1/B)}$ can be rewritten as

${P(A_1/B) = frac{P(A_1) times P (B/A_1)}{P(A_1)} times P
(B/A_1) + P (A_2) times P (BA_2)}$

Hence the general form of Bayes Theorem is

${P(A_i/B) = frac{P(A_i) times P (B/A_i)}{sum_{i=1}^k P(A_i)
times P (B/A_i)}}$

Where ${A_1}$, ${A_2}$…${A_i}$…${A_n}$ are set of n mutually
exclusive and exhaustive events.

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