Discrrete mathematics for computer science bayes

Motivating Example: Drug TestsA drug test gives a false positive 2% of the time that is, 2% of those who test positive actually are not drug users And the same test gives a false negativ

Bayes Theorem

Motivating Example: Drug Tests

A drug test gives a false positive 2% of the time (that is, 2% of

those who test positive actually are not drug users)

And the same test gives a false negative 1% of the time (that is, 1%

of those who test negative actually are drug users)

If Joe tests positive, what are the odds Joe is a drug user?

Insufficient information!

Suppose we know that 1% of the population uses the drug?

Setting Up the Drug Test Problem

• Let T be the set of people who test positive

• Let D be the set of drug users

• These are events, and Pr(T|D) is the probability that a drug user tests positive

• Pr(T|D) = 99 because the false negative rate is 1%, that is, 99%

of drug users test positive, 1% test negative

• We want to know: What is Pr(D|T)?

• This is a very different question: What is the probability you are

a drug user, given that you test positive?

Bayes Theorem

Theorem: If Pr(A) and Pr(B) are both nonzero,

Proof We know that

by the definition of conditional probability:

and similarly for Pr(B|A)

Then divide the left and right sides of (*) by

Pr(B|A)∙Pr(B)

Pr(A | B) Pr(B | A) = Πρ( Α )

Πρ( Β )

Pr(A | B) = Πρ( Α ∩ Β )

Πρ( Β ) ,

Bayes, v 2

This enables us to calculate Pr(A|B) using only the absolute probability Pr(A) and the conditional probabilities Pr(B|A) and Pr(B|¬A)

Proof We know that

Now multiply by Pr(B|A) and rewrite Pr(B) using the law of total

probability

Pr(A | B) = Πρ( Α )⋅Πρ( Β | Α )

Πρ( Β | Α )⋅Πρ( Α ) + Πρ( Β | Α )⋅Πρ( Α )

Pr(A | B) Pr(B | A) = Πρ( Α )

Πρ( Β )

Drug Test again

• Suppose that a drug test has

– 2% false positives (that is, 2% of the people who test positive are not drug users )

– 1% false negatives (1% of those who test negative are drug users)

• Suppose 1% of the population uses drugs If you test positive, what are the odds you are actually a drug user?

Drug test, cont’d

• Let D = “Uses drugs”

• Let T = “Tests positive”

• What is Pr(D|T)?

Pr(D) = 01

Πρ( Τ | ∆ ) = 02

Πρ( Τ | ∆ ) = 99

• If you fail the drug test, there is only one chance in three you are actually a drug user!

• How can this be? Think about it

– Out of 1000 people there are 10 drug users and 990 non-users

– Of those 990, 2% or almost 20 test positive

– Almost all of the 10 users also test positive

– So there are 2 non-users for every user, among those who test positive!

Pr(D | T ) = Πρ( ∆ )⋅Πρ( Τ | ∆ )

FINIS