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Bayes’ Theorem

Bayes' theorem converts one conditional into the other. You usually know P(evidence∣hypothesis)P(\text{evidence} \mid \text{hypothesis}), which is a property of your model, and you want P(hypothesis∣evidence)P(\text{hypothesis} \mid \text{evidence}), which is what you should now believe.

Bayes theorem
P(A∣B)=P(B∣A) P(A)P(B)P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}

It reverses the conditioning, which is what turns evidence about a cause into a belief about it.

Each piece has a name worth knowing:

  • P(A)P(A) is the prior: your belief before the evidence.
  • P(B∣A)P(B \mid A) is the likelihood: how well the hypothesis explains what you saw.
  • P(B)P(B) is the evidence: how likely the observation was overall, which normalises the result.
  • P(A∣B)P(A \mid B) is the posterior: your updated belief.

The denominator is almost always computed with the law of total probability:

P(B)=P(B∣A)P(A)+P(B∣Ac)P(Ac)P(B) = P(B \mid A)P(A) + P(B \mid A^{\mathsf{c}})P(A^{\mathsf{c}})

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