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How does Bayes's theorem calculate a cause from an observed result?

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sample space for a pair of dice — probability theory Sample space for a pair of dice.

Bayes’s theorem calculates the probability of a cause (or hypothesis) by working backward from an observed result (evidence). Often called the law of inverse probability, it allows for the revision of initial beliefs in light of new data.123

To calculate this, you assign a prior probability to a hypothesis, $P r (H)$ , representing your initial belief before observing the evidence. You also determine the conditional probabilities of the evidence occurring if the hypothesis is true, $P r_{H} (E)$ , and if it is false, $P r_{- H} (E)$ . The theorem then provides the posterior probability, $P r_{E} (H)$ , using the following formula:43

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

This process effectively updates the likelihood of the cause based on the strength of the observed evidence. While early applications were limited by the subjective nature of prior probabilities or complex calculations, modern computing and improved algorithms have enabled its widespread use in fields ranging from statistical inference to decision analysis.43