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(+) = P(+|AIDS)P(AIDS)/P(+) How do we find P(+) when we know P(+|AIDS), P(+|no AIDS)?

Day 8 Example 1

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)

Example 1: Governator Votes WhiteBlackHisp.OtherTotal Arnold A' Total P(W|A) = P(W  A)/P(A) =.364/.4522 =.805 >.7

Example 2: SPAM filters

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))

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

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”

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

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