Probability notation for Bayes' rule
These quantities are often written using conditional probabilities:
Prior belief in the hypothesis: \(\mathbb P(H).\)
Likelihood of evidence, conditional on the hypothesis: \(\mathbb P(e \mid H).\)
Posterior belief in hypothesis, after seeing evidence: \(\mathbb P(H \mid e).\)
For example, Bayes’ rule in the odds form describes the relative belief in a hypothesis \(H_1\) vs an alternative \(H_2,\) given a piece of evidence \(e,\) as follows:
- Probability notation for Bayes' rule: Intro (Math 1)
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Bayes’ rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation.