1. 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)
???
???

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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)

8. “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

9. “LAW OF TOTAL PROBABILITY”
BAYES’ FORMULA
Assume B1, B2, …, Bn “partition” the population, i.e., they are disjoint and exhaustive.
B B B ……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”

10. 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.
B B B ……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

11. 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+

12. 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

13. 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+) = ???

14. 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

15. The Debate Continues…
Frequentists Forever
Bayes’ Rules!

16. See...
EXAM 1 ENDS HERE