Bayes Theorem

A mathematical formula describing how to update the probability of a hypothesis based on new evidence, combining a prior probability with a likelihood.

Source: Bayes 1763

Bayes' theorem is a formula for revising the probability of a hypothesis in the light of new evidence, combining a prior probability, what was believed beforehand, with the likelihood of the evidence under that hypothesis. It updates belief by weighting the prior against how well the hypothesis explains the new data, producing a posterior probability that reflects both. In medicine this describes how a test result should shift the estimated chance a patient has a disease, given how common it is. Turning prior belief plus evidence into a revised estimate is its function. Bayes' theorem underlies Bayesian statistics, as Sutton and Abrams (2001) discuss.

Source: Sutton & Abrams 2001

Bayes' theorem works by combining the prior probability of a hypothesis with the likelihood of the observed evidence under that hypothesis, and normalising, to produce the posterior probability of the hypothesis given the evidence. In effect, the posterior is proportional to the prior multiplied by the likelihood. Stronger evidence, or a more informative likelihood, shifts the posterior further from the prior. So Bayes' theorem works by multiplying prior belief by the evidence's likelihood to update to a posterior belief, quantifying exactly how much a given piece of evidence should change the probability assigned to a hypothesis, which is the mechanism of Bayesian updating.

Source: Bayes 1763

Bayes' theorem is important because it provides the principled rule for updating beliefs with evidence, underpinning Bayesian statistics and inference, and it clarifies reasoning about conditional probabilities, such as interpreting diagnostic test results, where the probability of disease given a positive test depends on the prior probability. It corrects intuitions that neglect base rates. So Bayes' theorem matters as the foundation of Bayesian methods and as a guide to sound probabilistic reasoning, since it shows how evidence should be combined with prior knowledge, which is central in statistics, diagnosis, and decision-making, and helps avoid errors that arise from ignoring prior probabilities.

Source: Bayes 1763

Bayes' theorem is applied in interpreting diagnostic tests, where it combines the prior probability of disease with the test's performance to give the probability of disease given the result; in Bayesian statistics, where it updates prior distributions of parameters into posteriors using data; and in decision analysis and evidence synthesis. So Bayes' theorem is applied wherever beliefs or probabilities must be updated with evidence, from clinical diagnosis to statistical modelling, providing the formal basis for combining prior information with new data, which makes it fundamental to Bayesian approaches in health research and to the correct interpretation of tests and probabilities in medicine.

Source: Bayes 1763

In Bayes' theorem, the prior is the probability assigned to a hypothesis before observing the new evidence, representing existing belief or knowledge, and the posterior is the updated probability after incorporating the evidence, obtained by combining the prior with the likelihood. The prior is the starting point and the posterior the result of updating. So the prior and posterior are the before and after of Bayesian updating, with the theorem specifying how the evidence transforms one into the other, and the choice of prior influences the posterior, especially when the evidence is limited, which is why the selection of the prior matters in Bayesian analysis.

Source: Bayes 1763