P3. Independence and Bayes' Rule

Question 1

Notation for “A is independent of B”

Question 2

Condititions for events to be independent

Answer 2

1. P(A∩B) = P(A)P(B)
2. P(A|B) = P(A)

Note: symmetric (i.e. A is independent of B, iff B is independent of A).

Question 3

How do we derive the first condition for independence?

Answer 3

Question 4

How can we illustate independence on a Venn diagram?

Answer 4

The ratio of area A to Ω must equal to the ratio of area A∩B to B - thus knowing B does not help you learn anything about A.

Question 5

What is Bayes’ rule?

Answer 5

Question 6

How is Bayes’ rule derived?

Answer 6

Question 7

What is Beyesian learning?

Answer 7

Question 8

For:

What is the prior probability?

Answer 8

Question 9

For:

What is the likelihood?

Answer 9

Question 10

For:

What is the evidence?

Answer 10

Question 11

For:

What is the posterior probability?

Answer 11

Question 12

Suppose that 15% of cyclists dope. The blood test for doping is imperfect so: P(positive|dopes) = 0.9 P(positive|clean) = 0.12.

What is the probability that a cyclist dope given they test positive?

Answer 12