AHL 4.13—Bayes theorem

Bayes' theorem, named after 18th-century British mathematician Thomas Bayes, is one of the most powerful tools in probability theory. It provides a systematic way to update the probability of an event based on new evidence or information.

At its heart, Bayes' theorem formalises how we should rationally change our beliefs when we observe new data. This makes it invaluable in fields ranging from medicine and genetics to machine learning and forensic science.

Prior Probability: The probability of an event before new evidence is taken into account. Denoted $P(H)$, it represents our initial belief about how likely hypothesis $H$ is.

Posterior Probability: The probability of an event after updating with new evidence. For example, $P(H|E)$ is the posterior probability of hypothesis $H$ given that evidence $E$ has been observed.

Law of Total Probability: If events $H_1,H_2,\dots ,H_n$ are mutually exclusive and exhaustive (they form a partition of the sample space), then for any event $E$:
$P(E)={\sum }_{k=1}^{n}P(H_k)\cdot P(E|H_k)$
This is the foundation on which Bayes' theorem rests , it provides the denominator in every Bayes calculation.