posterior probability Bayes’ Formula Exactly how does one event A affect the probability of another event B? P(B) P(B) P(B ) A prior probability posterior probability P(B ∩ A) P(A) ??? ???

Consider the following scenario… Weather Forecast: P(Rain) = .50 prior probability Track Record: P(Overcast | Rain) = .60 P(Overcast | No Rain) = .20 Want: P(Rain | Overcast) = ??? posterior probability By def, P(Rain | Overcast) = Recall… .75 B Bc A Ac 1 Let A = Overcast, B = Rain Also recall (tree diagram) .75

Consider the following scenario… 1st column marginal Weather Forecast: P(Rain) = .50 prior probability Track Record: P(Overcast | Rain) = .60 P(Overcast | No Rain) = .20 Want: P(Rain | Overcast) = ??? posterior probability By def, P(Rain | Overcast) = Recall… .75 B Bc A Ac .5 1 Let A = Overcast, B = Rain Also recall (tree diagram) .75

Consider the following scenario… 1st column marginal Weather Forecast: P(Rain) = .50 prior probability Track Record: P(Overcast | Rain) = .60 P(Overcast | No Rain) = .20 Want: P(Rain | Overcast) = posterior probability By def, P(Rain | Overcast) = Recall… .75 B Bc A .3 .1 .4 Ac .5 1 Let A = Overcast, B = Rain Also recall (tree diagram) .75

Consider the following scenario… 1st column marginal Weather Forecast: P(Rain) = .50 prior probability Track Record: P(Overcast | Rain) = .60 P(Overcast | No Rain) = .20 Want: P(Rain | Not Overcast) = ??? posterior probability By def, P(Rain | Overcast) = Recall… Exercise B Bc A .3 .1 .4 Ac .2 .6 .5 1 Hint: Use probability table Also recall… Also recall (tree diagram) .75

Consider the following scenario… 1st column marginal Weather Forecast: P(Rain) = .70 prior probability Track Record: P(Overcast | Rain) = .60 P(Overcast | No Rain) = .20 Want: P(Rain | Overcast) = ??? posterior probability By def, P(Rain | Overcast) = Recall… B Bc A .42 .06 .48 Ac .28 .24 .52 .70 .30 1 Let A = Overcast, B = Rain Also recall (tree diagram) .75 .875

posterior probability Bayes’ Formula Exactly how does one event A affect the probability of another event B? P(B) P(B) P(B ) A prior probability posterior probability P(B ∩ A) P(A) Recall…… P(B | A) = P(A | B) P(B) P(A) Bayes’ Law (short form)

“LAW OF TOTAL PROBABILITY” Bayes’ Law (full form, n = 2) Bayes’ Law (short form) P(A | B) P(B) P(A) P(B | A) = ??? Recall… “LAW OF TOTAL PROBABILITY” Also recall…… For any two events A and B, there are 4 disjoint intersections: A ⋂ B A ⋂ Bc Ac ⋂ B “A only” “B only” Ac ⋂ Bc “Neither A nor B” “A and B” A B Bc Probability Table B BC A P(A ⋂ B) P(A⋂ BC) P(A) AC P(AC ⋂ B) P(AC ⋂ BC) P(AC) P(B) P(BC) 1.0

“LAW OF TOTAL PROBABILITY” BAYES’ FORMULA Assume B1, B2, …, Bn “partition” the population, i.e., they are disjoint and exhaustive. B1 B2 B3 ……etc……. Bn A Ac ……etc……. P(A) P(A ∩ B1) P(A ∩ B2) P(A ∩ B3) P(A ∩ Bn) Given… P(Ac ∩ B1) P(Ac ∩B2) P(Ac ∩ B3) ……etc……. P(Ac ∩ Bn) P(Ac) Prior probabilities: 1 P(B1) P(B2) P(B3) ……etc……. P(Bn) Conditional probabilities: P(A|B1) P(A|B2) P(A|B3) ……etc……. P(A|Bn) Then… Posterior probabilities: P(B1|A) P(B2|A) P(B3|A) ……etc……. P(Bn|A) are computed via P(Bi ∩ A) P(A) P(A | Bi) P(Bi) P(A | B1) P(B1) + P(A | B2) P(B2) + …+ P(A | Bn) P(Bn) = P(Bi | A) = P(A) = P(A | B1) P(B1) + P(A | B2) P(B2) + …+ P(A | Bn) P(Bn) for i = 1, 2, 3,…, n “LAW OF TOTAL PROBABILITY”

P(A | B1) P(B1) + P(A | B2) P(B2) + …+ P(A | Bn) P(Bn) BAYES’ FORMULA Assume B1, B2, …, Bn “partition” the population, i.e., they are disjoint and exhaustive. B1 B2 B3 ……etc……. Bn A Ac ……etc……. P(A) P(A ∩ B1) P(A ∩ B2) P(A ∩ B3) P(A ∩ Bn) Given… P(Ac ∩ B1) P(Ac ∩B2) P(Ac ∩ B3) ……etc……. P(Ac ∩ Bn) P(Ac) Prior probabilities: 1 P(B1) P(B2) P(B3) ……etc……. P(Bn) ? ? ? ? ……etc…… INTERPRET! Then… Posterior probabilities: P(B1|A) P(B2|A) P(B3|A) ……etc……. P(Bn|A) are computed via P(Bi ∩ A) P(A) P(A | Bi) P(Bi) P(A | B1) P(B1) + P(A | B2) P(B2) + …+ P(A | Bn) P(Bn) = P(Bi | A) = NOTE: If any prior P(Bi) = its posterior P(Bi | A), then that Bi and A are statistically independent! for i = 1, 2, 3,…, n

Example: Screening Tests: D = Disease (+ or –), T = Test (+ or –) “prevalence” prior probabilities P(D+) = .10 P(D–) = .90 T– T+ Ex: Colorectal Cancer T+ ∩ D+ T+ ∩ D– Gold Standard T– ∩ D+ T– ∩ D– D can be determined via a “gold standard” test… True Positive rate, “sensitivity” P(T+ | D+) = .85 False Positive rate P(T+ | D–) = .02 True Negative rate, “specificity” P(T– | D–) = .98 False Negative rate P(T– | D+) = .15 Highly sensitive and highly specific, but expensive. Cost-effective for adults 50+

Fecal Occult Blood Test (FOBT) Example: Screening Tests: D = Disease (+ or –), T = Test (+ or –) “prevalence” prior probabilities P(D+) = .10 P(D–) = .90 T– T+ Ex: Colorectal Cancer T+ ∩ D+ T+ ∩ D– T– ∩ D+ T– ∩ D– Fecal Occult Blood Test (FOBT) D can be determined via a “gold standard” test… True Positive rate, “sensitivity” P(T+ | D+) = .85 False Positive rate P(T+ | D–) = .02 True Negative rate, “specificity” P(T– | D–) = .98 False Negative rate P(T– | D+) = .15 Cheap, fast, easy Easy insurance coverage

posterior probabilities Example: Screening Tests: D = Disease (+ or –), T = Test (+ or –) “prevalence” prior probabilities P(D+) = .10 P(D–) = .90 T– posterior probabilities T+ Ex: Colorectal Cancer T+ ∩ D+ T+ ∩ D– T– ∩ D+ T– ∩ D– D can be determined via a “gold standard” test… True Positive rate, “sensitivity” P(T+ | D+) = .95 False Positive rate P(T+ | D–) = .45 True Negative rate, “specificity” P(T– | D–) = .55 True Negative rate, “specificity” P(T– | D–) = ??? False Negative rate P(T– | D+) = .05 False Negative rate P(T– | D+) = ???

Example: Screening Tests: D = Disease (+ or –), T = Test (+ or –) ??????????? “prevalence” prior probabilities P(D+) = .10 P(D–) = .90 T– posterior probabilities T+ “Predictive Value” (PV) Ex: Colorectal Cancer P(D+ | T+) T+ ∩ D+ T+ ∩ D– of a positive test: .19 T– ∩ D+ T– ∩ D– ??? P(D– | T–) of a negative test: .99 True Positive rate, “sensitivity” P(T+ | D+) = .95 False Positive rate P(T+ | D–) = .45 True Negative rate, “specificity” P(T– | D–) = .55 False Negative rate P(T– | D+) = .05

The Debate Continues… Frequentists Forever Bayes’ Rules!

See... http://pages.stat.wisc.edu/~ifischer/Intro_Stat/stat311/PowerPoint/Chapter_2/More_Bayes'_Examples EXAM 1 ENDS HERE