Lesson 5 of 5 · 14 min

Bayes in practice: diffuse priors, missing inputs and screening tests

Exam Bayes problems usually hide one input: you may need to back it out with the total probability rule, use complements, or start from equal (diffuse) priors.

In short

  • When P(B)P(B) is given directly, Bayes is one line: P(A∣B)=[P(B∣A)/P(B)]×P(A)P(A \mid B) = [P(B \mid A)/P(B)] \times P(A).
  • If a likelihood is missing, solve for it with the total probability rule.
  • Complements help: P(fail∣S)=1−P(pass∣S)P(\text{fail} \mid S) = 1 - P(\text{pass} \mid S) and P(fail)=1−P(pass)P(\text{fail}) = 1 - P(\text{pass}).
  • A diffuse prior gives every event equal probability, so the posterior is driven entirely by the likelihoods.
  • Information is useful when it moves the posterior well away from the prior.

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Bayes in practice: diffuse priors, missing inputs and screening tests · Probability Trees and Conditional Expectations