Bayes Thomas

Thomas Bayes is the scientist who formulated the Bayes theorem.Thomas Bayes is an English Nonconformist theologian and mathematician who was the first to use probability and who established a mathematical basis for probability inference (a means of calculating, from the frequency with which an event has occurred in prior trials, the probability that it will occur in future trials.)Bayes theorem describes the relationship between a conditional probability and the reverse of conditional probability.



Statement of Bayes Theorem Formula Theorem

Thomas Bayes gives a continuous prior and posterior probability distributions and discrete probability distributions of data, but the simplest setting involves only in discrete distributions. Bayes' theorem related to the conditional probability and marginal probabilities of events H and J.

Each term has a conventional name in Bayes Theorem:

P(H) is the prior probability or marginal probability of H.

P(H|J) is the conditional probability of  H, given J.

P(J|H) is the conditional probability of J, given H.

P(J) is the prior probability or marginal probability of J, P(J) acts as a normalizing constant.
Example - Bayes Thomas Theorem

Problem 1: In a hospital 2% of the women have cancer. A woman with the cancer has a 80% chance of a positive result in a test, while a woman without the cancer has a 20% chance of a false positive (negative) result. What is the probability of a woman has cancer given that she just had a positive result in a test?

Solution:       Consider H = "the woman has cancer" and A = "a positive test".

By using Bayes formula,

 P(A|H)* P(H) = 0.02 * 0.8 = 0.016

For finding the P(A) We need following details.

P(H1) -Probability of women having cancer

P(A|H1) -Probability for getting positive result given women having cancer

P(H2)  -Probability of women not having cancer

P(A|H!) -Probability for getting positive result given women not having cancer

So by using the multiplication rule.

= 0.02 x 0.8 + 0.98 x 0.2

= 0.016+0.196

= 0.212

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So the probability a woman has cancer given that she just had a positive test.

 P(H|A) = 0.016/0.212

= 16/212

= 0.0755