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How does Bayes’s theorem update a probability after new information is known?

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probability and statistics
Swiss commemorative stamp of mathematician Jakob Bernoulli, issued 1994, displaying the formula and the graph for the law of large numbers, first proved by Bernoulli in 1713.

Bayes’s theorem is a fundamental tool in probability theory used to update the probability of a hypothesis as new evidence becomes available.12

The process begins by assigning a "prior probability," denoted as $P r (H)$ , which represents the initial belief in a hypothesis $H$ before considering new data. When new evidence $E$ is obtained, the theorem incorporates the conditional probabilities of that evidence—specifically, the likelihood of the evidence given the hypothesis is true ( $P r_{H} (E)$ ) and the likelihood given the hypothesis is false ( $P r_{- H} (E)$ ).12

These values are combined using the following formula to calculate the "posterior probability," or the updated probability of the hypothesis given the evidence ( $P r_{E} (H)$ ):

$P r_{E} (H) = \frac{P r (H) P r_{H} (E)}{P r (H) P r_{H} (E) + P r (- H) P r_{- H} (E)}$

By applying this calculation, one can mathematically revise predictions. This method is central to Bayesian analysis, allowing researchers to refine their understanding of parameters or states of nature by systematically integrating new information with existing knowledge.21