Bayes

1
Bayes Theorem

• Bayes Theorem allows us to calculate the
  conditional probability one way (e.g., P(BA)
  when we know the conditional probability the
  other way (P(AB).

2
Bayes Theorem

• We use the following logic
• P(A n B) P(A) P(BA)
• P(A n B) P(B) P(AB)
• Thus, P(A) P(BA) P(B) P(AB)
• And P(BA) P(B) P(AB)
• P(A)

3
Bayes Theorem

• All that remains is to figure out P(A) for the
  denominator.
• In the simplest case, there are two ways A could
  happen A and B, or A and B.
• P(A) P(A n B) P(A n B)
• By the formula for conditional probability
• P(A) P(B)P(AB) P(B)P(AB)

4
Bayes theorem

• P(BA) P(B) P(AB)
• P(A)
• P(A) P(B)P(AB) P(B)P(AB)
• This leads us to Bayes Theorem
• P(BA) P(B) P(AB)
• P(B)P(AB) P(B)P(AB)

5
Bayes Theorem Example

• It is known that at any one time, one person in
  200 has a given disease. A new test for this
  disease is developed and a study of this tests
  reliability is performed, using samples of people
  known to have or known not to have the disease.
  Of 1000 people known to have the disease, 990
  test positive. Of 5000 people known not to have
  the disease, 250 test positive. What is the
  probability that a randomly-selected person from
  the general population has the disease, given
  that they test positive?

6
Bayes Theorem Example

• What we know from the question
• Ill Not Ill
• Test 990 (99) 250 (5)
• Test 10 (1) 4750 (95)
• 1000 5000

7
Bayes Theorem Example

• What is the probability that a randomly-selected
  person from the general population has the
  disease, given that they test positive?
• P(Ill) 1/200 .005 P(Not Ill) 1 .005
  .995
• P(T Ill) .99 P(TNot Ill) .05
• P(Ill T) P(Ill) P(TIll)
• P(Ill)P(TIll) P(Not Ill)
  P(TNot Ill)

8
Bayes Theorem Example

• P(IllT) (.005)(.99)
• (.005)(.099) (.995)(.05)
• .00495
• (.00495) (.04975)
• .0905

9
Bayes Theorem Example

• We can generalize this approach to cases where B
  consists of 3 or more mutually exclusive,
  exhaustive events
• P(BiA) P(Bi) P(ABi)
• P(B1)P(AB1)P(B2)P(AB2) P(Bk)P(ABk)