In the 2004 presidential election of those Texans who voted for either Kerry or Bush 62 voted for Bush and 38 for Kerry Of the Massachusetts residents who voted for either Kerry or Bush ID: 742708
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Slide1
Bayesian Notions and False PositivesSlide2
In the 2004, presidential election, of those Texans who voted for either Kerry or Bush,
62% voted for Bush and
38% for Kerry.
Of the Massachusetts residents who voted for either Kerry or Bush,
37% voted for Bush and
63% for Kerry.
Bill was a Kerry voter. He comes from either Texas or Massachusetts but I know nothing more about him.
Is it more likely that he comes from Texas or from Massachusetts?Slide3
I need to tell you that:
in Texas there were 7.4 million voters for either Kerry or Bush and
in Massachusetts there were only 2.9 million such voters.Slide4
I need to tell you that in Texas there were 7.4 million voters for either Kerry or Bush and in Massachusetts there were 2.9 million such voters.
Thus, of the Kerry voters from the two states, 61% came from Texas and only 39% came from Massachusetts.
Slide5
Thus, of the Kerry voters from the two states, 61% came from Texas and only 39% came from Massachusetts.
So Bill is more likely a Texan
.Slide6
MASS
.
TEXAS
VOTE BUSH
VOTE BUSH
VOTE KERRY
VOTE KERRYSlide7
DISEASE
.
NOT DISEASE
VOTE BUSH
VOTE BUSH
VOTE KERRY
VOTE KERRYSlide8
DISEASE
.
NOT DISEASE
TEST POSITIVE
TEST POSITIVE
TEST NEGATIVE
TEST POSITIVESlide9
Bayes’ Theorem
Where:
Is the probability of Event B given that Event A has occurred
Is the probability of Event A given that Event B has occurred
Is the probability of Event B
Is the probability of Event ASlide10
Bayes’
Theorem for
Kerry_voter
vs. TexanSlide11
False Positives in Medical Tests
Suppose that a test for a disease generates the following results:
if a tested patient has the disease, the test returns a positive result 99.9% of the time, or with probability 0.999
2. if a tested patient does not have the disease, the test returns a negative result 99.5% of the time, or with probability 0.995.
Suppose also that only 0.2% of the population has that disease, so that a randomly selected patient has a 0.002 prior probability of having the disease.Slide12
False Positives in Medical Tests
Suppose that a test for a disease generates the following results:
if a tested patient has the disease, the test returns a positive result 99.9% of the time, or with probability 0.999
2. if a tested patient does not have the disease, the test returns a negative result 99.5% of the time, or with probability 0.995.
Suppose also that only 0.2% of the population has that disease, so that a randomly selected patient has a 0.002 prior probability of having the disease.
What is the probability of a “false positive”:
The patient does not have the disease
given that the test was positive?Slide13
Let’s begin with
What is the probability of a “true positive”:
The patient does have the disease
given that the test was positive?
+:
Patient Tests Positively
D:
Patient Has Disease
Is the probability Patient Tests Positively given that Patient Has Disease
Is the probability Patient Has Disease
Is the probability Patient Tests Positively
Is the probability Patient Has Disease given that Patient Tests Positively Slide14
Let’s begin with
What is the probability of a “true positive”:
The patient does have the disease
given that the test was positive?
+:
Patient Tests Positively
D:
Patient Has Disease
Is the probability Patient Tests Positively given that Patient Has Disease
Is the probability Patient Has Disease
Is the probability Patient Tests Positively
Is the probability Patient Has Disease given that Patient Tests Positively Slide15
Let’s begin with
What is the probability of a “true positive”:
The patient does have the disease
given that the test was positive?
+:
Patient Tests Positively
D:
Patient Has Disease
.999
.002
Is the probability Patient Tests Positively
Is the probability Patient Has Disease given that Patient Tests Positively Slide16
What is the
probability that the Patient Tests Positively?
.999
.002
.006988
Is the probability Patient Has Disease given that Patient Tests Positively Slide17
.999
.002
.006988
.2859 and
P(not
D|+)
is 1-.2859 = .7141
+:
Patient Tests Positively
D:
Patient Has Disease
What is the probability of a “false positive”:
The patient does not have the disease
given that the test was positive?Slide18
DISEASE
NOT
DISEASE
TEST NEGATIVE
TEST POSITIVESlide19
TEST NEGATIVE
+
+
DISEASE
DISEASE
NOTSlide20
What if the test was more accurate for those who did not have the disease?Slide21
DISEASE
NOT
DISEASE
TEST NEGATIVE
TEST POSITIVE