ProblemSession3

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MS&E 252Decision Analysis IProblem Session 3

Announcements

• Make sure to CC your buddy in any emails to your TA

• Amount of time spent on homeworks

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What concepts do we expect you to master?

• Distinctions• Kind, degree

• Probability• Background state of

information (&)• Inferential notation

• Probabilistic inference• Tree flipping

• Relevance• Relevance Diagrams

• The Rules of Arrow Flipping

• Associative Logic Errors

What are the important distinctions about distinctions?

• Kind: type of grouping (beer drinker, college graduate)

• Degree: separation within grouping (beer drinker vs. not beer drinker)

Kinds Degrees

Probability tree representation:

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We encode our uncertainty using probability.

Uncertainty comes from our lack of knowledge.

• Probability allows us to “speak precisely about our ignorance.”

• Instead of “The probability is …” say “I assign a probability … to …”

Your probability changes as your knowledge changes.

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Probability Notation

� your background state of information

� probability of event A occurring given your background state of information

� probability of event A occurring given you know event B occurred and your background state of information

� probability of event AB occurring given your background state of information

&&

{A|&}{A|&}

{A|B,&}{A|B,&}

{AB|&}{AB|&}

means{A | B, &} = 0.8“Given B and my background state of information, I assign a

probability of 0.8 to A”

We will use Inferential Notation for probabilistic statements.

• Two remarks:– We condition A on B when we think about A given

that B happened.– Always condition probabilities on & (your

background state of information)

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Probabilistic inference is how we learn about uncertainties indirectly.

We start with assessed probabilities and ...

… we can come up with probabilities we have not assessed.

… through probabilisticinference ...

Tree-flipping

But what exactly is tree-flipping?

In this example, we first calculate the joint distribution of two distinctions.

{A | &} {B | A,&} {AB | &} = {A | &} * {B | A,&}

A1 B1

A1 B2

A2 B1

A2 B2

0.9

0.1

0.4

0.6

0.5

0.5

0.36 = {A1 B1 | &} = (0.9)*(0.4)

0.54

0.05

0.05

Σ = 1.00

A1

A2

&

B1

B1

B2

B2

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We can then “flip” the tree to infer the unassessed probabilities.

{B | &} {A | B,&} {BA | &} = {B | &} * {A | B,&}

B1 A2

B2 A1

0.41

0.59

B1A1 0.36 {A1 B1 | &} = {B1 A1 | &}

B2 A2 0.05

Σ = 1.00

0.05

0.54

B1

B2

&

A1

A1

A2

A2

0.360.41 = 0.88

0.050.41 = 0.12

0.540.59 = 0.92

0.050.59 = 0.08

Here is an example of probabilistic inference.

• I have two coins in my pocket; one is normal (N) and the other is “double-headed” (D).

• I take a coin out of my pocket and flip it – “Heads”!• What is the probability that I originally chose the double-

headed coin?

0.25

1/3

2/3

0.25 = “H”N

0.50 = “H”D

0.25 = “T”N

0.25 = “H”N

0.25 = “T”N

0.50 = “H”D

0 = “T”D

0.75

D

“H”

“T”

N

N

0.5

0.5

N

D

0.5

0.5

“H”

“T”

“H”

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Let’s flip this tree as an exercise!

3/5

2/5

7/10

3/10

1/10

9/10

A

~A

&

B

B

~B

~B

23/50

27/50

21/23

2/23

1/3

2/3

21/50

9/50

2/50

18/50

&

B

~B

A

A

~A

~A

21/50

2/50

9/50

18/50

This tree is a little trickier.

3/50

12/50

15/50

10/50

4/50

6/50

D

D’

C1

C3

C2

C1

C3

C2

3/22

4/22

15/22

3/28

6/28

10/28

3/50

4/50

15/50

12/50

6/50

10/50

22/50

28/50

3/10C1

1/5

4/5

D

D’

3/5

2/5

D

D’

2/5

3/5

D

D’

2/10C2

5/10C3

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Introducing Relevance

Probabilistically, A is relevant to B if {A|B, &} is not equal to {A|B’, &}.

• In other words, if knowing B tells you something about the probability of A occurring, then A is relevant to B.

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Introducing Relevance

• Relevance, like probability, describes a person’s beliefs about the world, not the world itself.

• Relevance does not imply causality.• Relevance is a matter of information,

not logic.� A and B could be relevant given &,

and yet irrelevant given C and &.

Probability trees can help determine whether distinctions are “relevant” to each other.

• If knowing outcome A tells you something about the probability of B, then A is relevant to B.

• Otherwise they are irrelevant.• A is relevant to B iff {B|A, &} ≠ {B|~A, &}.

~B

1/3

2/3

3/4

1/4

3/4

1/4

A

~A

B

B

~B

~B

1/3

2/3

3/4

1/4

1/10

9/10

A

~A

B

B

~B

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0.1

How can we recognize relevance using trees? (3 or more degrees)

A and B are relevant given & if ...

Otherwise, A and B are irrelevant given &. A1

A3

B1

0.2

0.5

0.3B2

B3

B1

0.2

0.5

0.3B2

B3

B1

0.8

0.1B2

B3

A2 }}

… for one of the degrees of A, the distribution of B differs from ...

… the distribution of B for another degree of A.

How can we recognize relevance using trees? (3 or more distinctions)

B and C are relevant given distinction A and & if ...

Otherwise, B and C are irrelevant given distinction A and &.

A1

A2

B1

B1

B2

B2

C1

C2

C1

C2

C1

C2

C1

C2

}… either this distribution differs from...

} … this one ...

-- OR --

} … this distribution differs from...

} … this one.

0.3

0.3

0.3

0.6

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But sometimes we first need to flip the tree to determine whether there is relevance.

0.6

0.4

0.5

0.5

0.6

0.40.250.75

0.50.5

0.50.5

0.70.3

A B C

Are A and B irrelevant given C and &?

C A B

Our strategy is to put C and & first in the tree, then put A and B, and then put everything else.

… this differs from ...

… this

… this differs from ...

… this.

-- OR --

Then look at the tree. A and B are relevant given C if ...

One problem with treesis that they grow exponentially.

It would thus be nice to have another tool that would help us reflect on relevance regardless of the size of the tree.

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Introducing Relevance Diagrams

Relevance diagrams can make irrelevance statements…

… but they cannot make any relevance statements!

You should get used to saying “the diagram shows that there is a possibility of relevance between A and B”.

Clarifying our notation...

UncertaintyNote that these are kinds, not degrees of the uncertain distinction!

Arrows indicate the possibility of relevance.

So when building or assessing a relevance diagram, the biggest statements made are those of irrelevance!

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Instead of trees, we can relevance diagrams.

{A | &} {B | A, &}

No arrowspointing in

Arrow pointing from A into B –

conditions B on A

A1

A2

B1

B1

B2

0.626

0.374

0.714

0.286

0.552

0.448

B2

A B

The arrow from A to B only implies possible relevance.

Probability Tree Relevance Diagram

Relevance diagrams allow us to make strong statements of irrelevance between distinctions.

A1

A2

B1

B1

B2

0.626

0.374

0.714

0.286

0.714

0.286

B2

The absence of an arrow from A to B asserts irrelevance!

Probability Tree Relevance Diagram

A B

{AB|&} = {A|&}{B|&}

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Diagrams vs. Trees

B C

A

A is IRR to C | B, &

From trees we can make both relevance and irrelevance

statements.

A1

A2

B1

B1

B2

B2

C1

C2C1

C2

C1

C2C1

C2

0.3

0.3

0.3

0.3

These two numbers are the same...

...these two numbers are the same

-- AND --

From diagrams we can onlymake statements of

irrelevance!

Example

– Assume you have two “fair” dice.– You believe the result of each die toss is

irrelevant to the other.

Die 1 Die 2

Relevance is a matter of information, not logic.

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But once you know the sum of the two tosses, they are now relevant to each other.

Adding or taking away information can change relevance relationships.

“The two tosses MAY BE relevant to each other given their sum and &.”

Die 1 Die 2

Sum

Die 1 Die 2

How are these two tools used?

B C

A

BB CC

AA• We start building the relevance diagram, as a way to clarify our thoughts and learn which assessments need to be made.

A1

A2

B1

B1

B2

B2

C1

C2

C1

C2

C1

C2

C1

C2

0.3

0.3

0.3

0.3

A1

A2

B1

B1

B2

B2

C1

C2

C1

C2

C1

C2

C1

C2

0.3

0.3

0.3

0.3

• We then make the necessary probability assessments and build our trees.� Note that it is the irrelevance statements

that reduce the number of probability assessments that need to be made.

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Irrelevance helps us simplify our probabilistic thinking.

B C

A

A | &

B | A, & C | B, &

Bayes’ Rule tells us that:{ABC|&}={A|&} * {B|A, &} * {C|B, A, &}

This diagram is telling us the same thing!{ABC|&} = {A|&}*{B|A, &}*{C|B, &}









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Just like we can flip trees, we can flip arrows in relevance diagrams.

A B

{A | &} {B | A, &}

A B

{A | B, &} {B | &}

Arrow flipping requires that the two distinctions be conditioned on the same state of information.

“Add arrows wherever you want, provided you don’t create a cycle; A cycle made by more than

3 nodes is also not allowed.”

A B

C

We can only flip arrows according to certain rules.

RULE #1

X X

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“You can flip an arrow between A and B if and only if A and B are conditioned on the same state

of information.”


RULE #2

In other words, any other node (here, C and D) which points to A also points to B. A B

D

C

?

*Tip* – Draw a box around A and B

“You cannot remove any arrows arbitrarily.”


RULE #3

A B

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Example of diagram manipulation(arrow-flipping)

D E

BA

CCan we flip the arrow between A and B?

No, since A and B are conditioned on different states of information!� A has arrows from C and D, but B has arrows from D and E.

Q

A

D E

BA

C

We need to add arrows from C to B and E to A.

Now, A and B are conditioned on the same state of information, so we can now flip the arrow.

Example of diagram manipulation(arrow-flipping)

A

A

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Why would we want to manipulate diagrams?

• We can recognize irrelevance without needing to assess numbers.

• If there is no arrow between nodes A and B given the same state of information S, then A and B are irrelevant given S.

A B

� A and B are irrelevant given &

A B

C

� A and B are irrelevant given C and &

An example of recognizing irrelevance from diagrams.

D E

BA

CConsider the following diagram:

• Are C and D irrelevant given &?� Yes; they are both conditioned only on & and there is no

arrow between them.

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D E

BA

CConsider the following diagram:

• Are C and D irrelevant given &?� Yes; they are both conditioned only on & and there is no

arrow between them.

• Are A and B irrelevant given C, D, E, and &?� Yes; add arrows from C to B and E to A.

• Are A and B irrelevant given &?

D E

BA

C

next line

D E

BA

C D E

BA

C

D E

BA

C D E

BA

C


So we can’t conclude! There is a possibility of relevance.

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Another example of recognizing irrelevance from diagrams.

Market sizeAnnual growth Cost

Competition Revenue Profit

• Cost and Competition irrelevant given &?• Revenue and Cost irrelevant given Market Size, &?• Profit and Annual Growth irrelevant given &?









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What is constitutes an Associative Logic Error?

We say that people make an “Associative Logic Error” when they fall into the trap

{A|B,&} = {B|A,&}

Examples of Associative Logic Errors

Some can be quite obvious…

Hemophiliacs are virtually all male…

… but the probability for a male to be a hemophiliac is

1/1000 or less!

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Examples of Associative Logic Errors

… But other examples can be very tricky.

Most people who are treated for lung cancer seem to be

heavy smokers…

… and yet the probability of getting lung cancer if you are a heavy smoker is as low as 0.1!

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