Probabilistic Theory of AI

Question 1

Give the definition of a probability distribution

Answer 1

function mapping a proposition -> [0,1]
K1: If a, p(a) = 1
K2: If !(a and b), p (a or b) = p(a) + p(b)

Question 2

How would you prove p(!a) = 1 - p(a)

Answer 2

we know (!a or a) = 1
we know !(!a and a), so p(!a) + p(a) = p(!a or a)
hence p(a) + p(!a) = 1 and p(!a) = 1 - p(a)

Question 3

How would you prove if a implies b then p(a) <= p(b)

Answer 3

we know !(a and !b)
(a or !b) = p(a) + p(!b) = p(a) + 1 - p(b)
hence p(b) - p(a) = 1 - p(a or !b) >= 0

Question 4

How would you prove if a implies b and b implies a then p(a) = p(b)

Answer 4

Prove p(a) <= p(b) and p(b) <= p(a)

Question 5

How would you prove p(a or b) = ?

Answer 5

p(a) + p(b) - p(a and b)

Question 6

A set of events (a, b) are mutually exclusive if?

Answer 6

p(a and b) = 0

Therefore, p(a or b) = p(a) + p(b)

Question 7

A set of events (a,b) are jointly exhaustive if?

Answer 7

p(a or b) = 1

Question 8

A set of events (a,b) form a partition if?

Answer 8

They are mutually exclusive and jointly exhaustive
Therefore p(a) + p(b) = 1

Question 9

How would you prove that if a set of events is mutually exclusive and forms a partition then p(e1) + … + p(en) = 1?

Answer 9

Induction on n

Question 10

Define conditional probability

Answer 10

p(a | b) = p(a and b) / p(b)

Question 11

Properties of conditional probability? Can you prove?

Answer 11

0 <= p(a|b) <= 1
if p(a) = 0, p (a | b) = 0
if b implies a, p(a | b) = 1
if b implies c and c implies b, p(a | b) = p(a | c)

Question 12

if we declare p_v(o) = p(o | v) what do we need to show

Answer 12

K1: if o, p(o | v) = 1
K2: if !(o and c), p(o or c | v) = p(o | v) + p(c | v)

Question 13

(p_a)_b(c) = ? = ?

Answer 13

p_(a and b) (c) = (p_b)_a (c)

Question 14

Law of total probability, p(a) = ?

Answer 14

p(a) = p(a | v1)p(v1) + … + p(a | vn)p(vn)

Question 15

Extend the law of total probability to p(a | b)

Answer 15

p(a | b) = p(a | b and v1)p(v1 | b) + … + p(a | b and vn)p(vn | b)

Question 16

Bayes Theorem, p(a | b)

Answer 16

p(a | b) = p(b | a)p(a) / p(b)