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Bayes’ Theorem

Fall 2014 EAS 305 Lecture Notes

Prof. Jun Zhuang

University at Buﬀalo, State University of New York

September 10, ... 2014

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 1 of 26

Conditional Probability

Bayes’ Theorem

Agenda

1

Conditional Probability

Deﬁnition and Properties

Independence

General Deﬁnition

2

Bayes’ Theorem

Partition

Theorem

Examples

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 2 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Example

Example: Die. A = {2, 4, 6}, B = {1, 2, 3, 4, 5}. So Pr(A) = 1/2, Pr(B) = 5/6.

Suppose we know that B occurs. Then the prob of A “given” B is Pr(A|B) =

|A ∩ B|

2

=

5

|B|

So the prob of A depends on the info that you have! The info that B occurs allows us to regard B as a new, restricted sample space. And. . .

Pr(A|B) =

|A ∩ B|

|A ∩ B|/|S|

Pr(A ∩ B)

=

=

.

|B|

|B|/|S|

Pr(B)

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 3 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Deﬁnition: If Pr(B) > 0, the conditional prob of A given B is Pr(A|B) ≡ Pr(A ∩ B)/Pr(B).

Remarks: If A and B are disjoint, then Pr(A|B) = 0. (If B

occurs, there’s no chance that A can also occur.)

What happens if Pr(B) = 0? Don’t worry! In this case, makes no sense to consider Pr(A|B).

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 4 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Example: Toss 2 dice and take the sum.

A: odd toss = {3, 5, 7, 9, 11}

B: {2, 3}

Pr(A) = Pr(3) + · · · + Pr(11) =

4

2

1

2

+

+ ··· +

= .

36 36

36

2

1

2

1

+

=

.

36 36

12

Pr(A ∩ B)

Pr(3)

2/36

Pr(A|B) =

=

=

= 2/3.

Pr(B)

Pr(B)

1/12

Pr(B) =

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 5 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Properties — analogous to Axioms of probability

1

0 ≤ Pr(A|B) ≤ 1.

2

Pr(S|B) = 1.

3

A1 ∩ A2 = ∅ ⇒ Pr(A1 ∪ A2 |B) = Pr(A1 |B) + Pr(A2 |B).

4

If A1 , A2 , . . . are all disjoint, then

∞

Pr

∞

Ai B

i=1

Prof. Jun Zhuang

Pr(Ai |B).

=

i=1

Fall 2014 EAS 305 Lecture Notes

Page 6 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Independence — Any unrelated events are independent

Deﬁnition: A and B are independent iﬀ

Pr(A ∩ B) = Pr(A)Pr(B).

A: It rains on Mars tomorrow.

B: Coin lands on H.

Example: If Pr(rains on Mars) = 0.2 and Pr(H) = 0.5, then

Pr(rains and H) = 0.1.

Note: If Pr(A) = 0, then A is indep of any other event.

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 7 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Remark: Events don’t have to be physically unrelated to be indep. Example: Die. A = {2, 4, 6}, B = {1, 2, 3, 4}, A ∩ B = {2, 4}, so Pr(A) = 1/2, Pr(B) = 2/3, Pr(A ∩ B) = 1/3.

Pr(A)Pr(B) = 1/3 = Pr(A ∩ B) ⇒ A, B indep.

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 8 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

More natural interpretation of independence

Theorem

Suppose Pr(B) > 0. Then A and B are indep iﬀ Pr(A|B) = Pr(A). Proof: A, B indep ⇔ Pr(A ∩ B) = Pr(A)Pr(B) ⇔

Pr(A ∩ B)/Pr(B) = Pr(A).

Remark: So if A and B are indep, the prob of A doesn’t depend on whether or not B occurs.

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 9 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Theorem

¯

A, B indep ⇒ A, B indep.

¯

Proof: Pr(A) = Pr(A ∩ B) + Pr(A ∩ B), so that

¯...