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Stat 321 – Day 9

Stat 321 – Day 9. Bayes’ Rule. Last Time. Multiplication Rule P(A  B) = P(A|B)P(B) or P(B|A)P(A) If the events are independent, simplifies to P(A  B) = P(A)P(B) Can use this relationship to numerically check for independence AIDS Problem P(AIDS|+) = P(+  AIDS)/P(+)

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Stat 321 – Day 9

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  1. Stat 321 – Day 9 Bayes’ Rule

  2. Last Time • Multiplication Rule • P(A  B) = P(A|B)P(B) or P(B|A)P(A) • If the events are independent, simplifies to P(A  B) = P(A)P(B) • Can use this relationship to numerically check for independence • AIDS Problem • P(AIDS|+) = P(+  AIDS)/P(+) = P(+|AIDS)P(AIDS)/P(+) How do we find P(+) when we know P(+|AIDS), P(+|no AIDS)?

  3. Day 8 Example 1

  4. Day 9 Example 1 • Slightly different question, given I have an Arnold supporter, what is the probability the person is white? • P(white|A) = P(white  A)/P(A) Law of Total Probability P(A|white)P(white)

  5. Example 1: Governator Votes .364 .0102 .0558 .0222 .4522 P(W|A) = P(W  A)/P(A) = .364/.4522 = .805 > .7 .70 .06 .18 .06 1.00

  6. Example 2: SPAM filters

  7. Example 3: Shadyside case • Defendant has same genetic markers and only .32% of male population has these markers, how would you “update” the probability of guilt for this defendant? • Want P(G|E) • Know P(E|G) = 1, P(E|G’) = .0032 • P(G|E) = P(G)/[P(G)+.0032(1-P(G))

  8. Example 2: Randomized Response • Technique for asking sensitive questions • Randomly decide which question respondents will answer: sensitive or boring • Work backwards with probability rules to estimate proportions for sensitive question

  9. Example 2: Randomized Response • Flip fair coin • Heads: answer sensitive question • Tails: answer boring question=“does your home phone number end in even digit?” • Determine proportion of “yeses” • Define events • Y=“response is yes” • S=“respondent answered sensitive question”

  10. Example 2: Randomized Response • Respondents are ensured confidentiality • Can still obtain estimate for P(Y|S)

  11. For Monday • HW 3 due Tuesday • Check out review sheet online this weekend • (Today’s handout – Day 9 - online has a Ch. 2 summary)

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