Bayes’ theorem calculates updated probability of a hypothesis given new empirical data and prior probabilities. Bayesian inference is its scientific application. The theorem is expressed as a formula and can be used to determine updated probability of any hypothesis given experimental results.
Bayes’ theorem, sometimes called Bayes’ rule or principle of inverse probability, is a mathematical theorem that follows very quickly from the axioms of probability theory. In practice, it is used to calculate the updated probability of some target phenomenon or hypothesis H given new empirical data X and some background information, or prior probabilities.
The prior probability of some hypotheses is usually represented by a percentage between 0% and 100%, or a number between 0 and 1. This probability is often called the degree of confidence, and is bound to vary from observer to observer, since not all observers have had the same experience and therefore cannot make equivalent probability estimates for any given hypothesis. The application of Bayes’ theorem in a scientific context is called Bayesian inference, which is a quantitative formalization of the scientific method. It allows for optimal review of theoretical probability distributions given experimental results.
Bayes’ theorem in the context of scientific inference says the following: “The new probability that a hypothesis H is true (called the posterior probability) given new evidence X is equal to the probability that we would observe this evidence X given that H is actually true (called the conditional probability, or likelihood), multiplied by the prior probability that H is true, all divided by the probability of X.”
A common restatement of the above in terms of how a test result contributes to the likelihood that a given patient has cancer can be shown as follows:
p(positive|cancer)*p(cancer)
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p(positive|cancer)*p(cancer) + p(positive|~cancer)*p(~cancer)
The vertical bar means “data”. The probability that the patient has cancer after a positive result on a given cancer test is equivalent to the probability of a positive result given cancer (derived from past results) times the prior probability that a given person has cancer (relatively low) all divided by the same number, plus the probability of a false positive multiplied by the previous probability of not having cancer.
Sounds complicated, but the above equation can be used to determine the updated probability of any hypothesis given any quantifiable experimental result.
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