Bayes′ Theorem
Bayes’ theorem:
Bayes' solution to a problem of "inverse probability" presented in the Essay Towards Solving a Problem in the Doctrine of Chances read after Bayes's death by Richard Price to the Royal Society in 1763, then published in the Philosophical Transactions of the Royal Society of London the following year.
Rearranges those two equations, we got:
Alternative form of the Bayes′ Theorem
If Ac is the complementary event of A (often called "not A"), we got:
Alternative form of the Bayes′ Theorem
About Thomas Bayes
1. Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures (1731) 2. An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst (published anonymously in 1736), in which he defended the logical foundation of Isaac Newton's calculus ("fluxions") against the criticism of George Berkeley, author of The Analyst.
Drug testing example
We need to compute:
Drug testing example
P(D), or the probability that the employee is a drug user, regardless of any other information. This is 0.005, since 0.5% of the employees are drug users. This is the prior probability of D. P(N), or the probability that the employee is not a drug user. This is 1 − P(D), or 0.995. P(+|D), or the probability that the test is positive, given that the employee is a drug user. This is 0.99, since the test is 99% accurate.
“Inverse probability”
In the first decades of the eighteenth century, many problems concerning the probability of certain events, given specified conditions, were solved. e.g. given a specified number of white and black balls in a box, what is the probability of drawing a black ball? Inverse probability: given that one or more balls has been drawn, what can be said about the number of white and black balls in the box?
Therefore the chance that a random trouser-wearer is a girl equals 20/80 = 0.25.
Drug testing example
Suppose a certain drug test is 99% sensitive and 99% specific, that is, the test will correctly identify a drug user as testing positive 99% of the time, and will correctly identify a non-user as testing negative 99% of the time. This would seem to be a relatively accurate test, but Bayes' theorem can be used to demonstrate the relatively high probability of misclassifying non-users as users.
Solve the problem with Bayes′ Theorem
Therefore the probability of seeing a student wearing trousers being a girl is 0.25.
Validation of Bayes′ Theorem
Bayes′ Theorem
P(B|A) is the conditional probability of B, given A. It is also called the likelihood. P(A) is the prior probability (or “unconditional” or “marginal” probability) of A. It is "prior" in the sense that it does not take into account any information about B; however, the event B need not occur after event A. P(B) is the prior or marginal probability of B. P(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.
Bayesian network and systems biology
王 秀 杰 xjwang@
Thomas Bayes (pronounced: beiz)
An English mathematician and presbyterian minister. In 1719 he enrolled at the University of Edinburgh to study logic and theology.
Solve the problem with Bayes′ Theorem
Event A: the student observed is a girl. Event B: the student observed is wearing trousers. What to compute?
Solve the problem with Bayes′ Theorem
Bayes′ Theorem
Bayes' theorem gives a mathematical representation of how the conditional probability of event A given B is related to the converse conditional probability of B given A.
Given a partition, i.e. {Ai}, of the event space, then:
Alternative form of the Bayes′ Theorem
When cover more than two events:
A simple example of Bayes' theorem
Drug testing example
Let's assume a corporation decides to test its employees for drug use, and that only 0.5% of the employees actually use the drug. What is the probability that, given a positive drug test, an employee is actually a drug user? Let "D" stand for being a drug user, "N" indicate being a non-user. Let "+" be the event of a positive drug test.