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10.3: Appendix

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  • Proof (Derivation of Bayes’ rule). First, remember the rule for computing a conditional probability:

    P(A|B)=P(AB)P(B) P(A|B) = \frac{P(A \cap B)}{P(B)}

    We can rearrange this to get the formula to compute the joint probability using the conditional:

    P(AB)=P(A|B)*P(B) P(A \cap B) = P(A|B) * P(B)

    Using this we can compute the inverse probability:

    P(B|A)=P(AB)P(A)=P(A|B)*P(B)P(A) P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{P(A|B)*P(B)}{P(A)}

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