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Bayesian Notions and False PositivesPowerPoint Presentation

Bayesian Notions and False Positives

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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?

I need to tell you that: voted for either Kerry or Bush,

in Texas there were 7.4 million voters for either Kerry or Bush and

in Massachusetts there were only 2.9 million such voters.

- 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.

- 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.

Bayes’ Theorem Texas and only 39% came from Massachusetts.

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 A

Bayes’ Texas and only 39% came from Massachusetts. Theorem for Kerry_voter vs. Texan

False Positives in Medical Tests Texas and only 39% came from Massachusetts.

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.

False Positives in Medical Tests Texas and only 39% came from Massachusetts.

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?

Let’s begin with Texas and only 39% came from Massachusetts.

What is the probability of a “true positive”:

The patient does have the disease

given that the test was positive?

A: Patient Tests Positively

B: Patient Has Disease

Is the probability Patient Has Disease given that Patient Tests Positively

Is the probability Patient Tests Positively given that Patient Has Disease

Is the probability Patient Has Disease

Is the probability Patient Tests Positively

Let’s begin with Texas and only 39% came from Massachusetts.

What is the probability of a “true positive”:

The patient does have the disease

given that the test was positive?

A: Patient Tests Positively

B: Patient Has Disease

Is the probability Patient Has Disease given that Patient Tests Positively

Is the probability Patient Tests Positively given that Patient Has Disease

Is the probability Patient Has Disease

Is the probability Patient Tests Positively

Let’s begin with Texas and only 39% came from Massachusetts.

What is the probability of a “true positive”:

The patient does have the disease

given that the test was positive?

A: Patient Tests Positively

B: Patient Has Disease

Is the probability Patient Has Disease given that Patient Tests Positively

.999

.002

Is the probability Patient Tests Positively

What is the Texas and only 39% came from Massachusetts.

probability that the Patient Tests Positively?

Is the probability Patient Has Disease given that Patient Tests Positively

.999

.002

.006988

What is the probability of a “false positive”: Texas and only 39% came from Massachusetts.

The patient does not have the disease

given that the test was positive?

A: Patient Tests Positively

B: Patient Has Disease

.2859 and P(not B|A) is 1-.2859 = .7141

.999

.002

.006988

What if the test was more accurate for those who did not have the disease?

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