1 Baysian networks for decision analysis The TV show problem as a - - PDF document

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Decision graphs I Decisions and utilities

Anders Ringgaard Kristensen

Several names

Decision graphs

• Decision trees
• Influence diagrams
• A certain kind of strictly symmetric decision trees
• “No forgetting” assumption
• LIMIDs – Limited Memory Influence Diagrams
• Influence diagrams without the “no forgetting”

assumption

Very often the terms “Decision graphs” and “Influence diagrams” are used synonymously.

Where are we?

First processing: Monitoring & filtering Second processing: Decision making Some methods integrate the whole setup

Bayesian Networks Decision Graphs

Bayesian networks to Decision graphs

If we have a Bayesian network and add:

• Decision nodes
• Utility nodes

Then we have a decision graph (if we obey certain rules) Algorithms for optimization of decisions are available

Notation, variables (= nodes)

C

Random variable, Chance node

D

Decision variable, Decision node D =

U

= U Utility variable, Utility node

Numerical contents Edges into a chance node (yellow circle) correspond to a set of conditional probabilities. They express the influence of the values of the parents on the value of the child. Edges into a utility node correspond to a function depending on the values of the parents. Edges into a decision node just means that the values of the parents are known when the decision is made. They are called information edges. The decision may depend on the values of its parents.

Parent 1 Child Parent 2 Parent 1 Child Parent 2 Parent 1 Child Parent 2

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The TV show problem as a BN

Refer do demo file ShowBN.xbn

Opened True Choice 1 Choice 2 Gain

Baysian networks for decision analysis

Problems

• More than one decision
• Many combinations to test
• The same decision is not necessarily the best

under all circumstances

• Complex strategies
• Cannot be entered as evidence
• Need for direct optimization

Exercise

Use the notation on Slide 5 and the rules on Slide 6 to change the TV show problem on Slide 7 to a decision

• graph. Consider:
• Which nodes are actually decision nodes (from the

player’s point of view)?

• Should we add any information edges?

Two kinds of decisions

Test decisions

• The decision to
• bserve the value of

a variable.

Actions

• Influences the value
• f at least one

variable.

• In this case:
• Fever
• Tired
• No influence on Flu

Flu Fever Tired

Test

Aspirin

Causal influences

The causal direction

Flu Fever Tired Flu Fever Tired

The reasoning direction

As long as we don’t include actions, the two Bayesian networks are equivalent If we include the action “Aspirin”, it would cure the flu in the latter case! In particular with decision graphs:

• Be careful with

causality!

How to model a test decision

Include a “mirror variable” for the observed node:

• The “mirror” has the same states AND a “not observed” state.

Flu Fever Tired Test Obs. fever

{No, yes} {Normal, medium, high} {No, yes} {No, yes} {Not observed, normal, medium, high}

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P(Obs. Fever | Fever, Test)

P(Obs. Fever | Fever, Test)

1 Medium 1 High Yes Medium 1 Yes Normal Normal Not observed Test Fever 1 No High 1 No Medium 1 No Normal Yes High

How to model the action

Include an “after-action” variable to capture the effect of the action.

• The “after-action” variable typically has the same states as the original

variable.

Flu Fever Test Obs. fever

{No, yes} {Normal, medium, high} {No, yes} {No, yes} {Not observed, normal, medium, high}

Tired Fever’ Aspirin

{No, yes} {Normal, medium, high}

Does it pay to perform tests?

What is the benefit?

• Costs of test.
• Improvement in “result”

Will the result change your actions?

• Are there any intervening options?
• Can you trust the result of the test?

What is the benefit?

Fever example:

Flu Fever Test Obs. fever Tired Fever’ Aspirin

Benefit Cost: Inconvenience Cost: Price of aspirin

Inconv Price

What is the total benefit? Consequences:

• Less tired
• Inconvenience of fever test
• Price of aspirin

Not comparable – cannot be added! We need a common unit for measurement The utility concept – a well known theory

Multi attribute utility: two attributes

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Utility concept in decision graphs

Decision graphs don’t support varying marginal rates of substitution between attributes. The values of all utility nodes of the graph are just summed and maximized. All utilities must be expressed in the same unit (e.g. money). No time preference allowed.

Constant marginal rate of substitution

Will the result change your actions?

The fever example:

• The aspirin action is the obvious and natural consequence of

the test.

• But, what about just taking an aspirin (without testing)?
• Inconvenience of test
• Cost of aspirin
• “You are not allowed to do that” – if true, a utility node

is missing!

• Reliability of test

Will the result change your actions?

The simple milk test example (example from Tuesday):

• The obvious action is not to pour the milk from a positive

test into the bulk tank.

• We would never do that unless we think the milk is infected.
• But, what about just pouring the milk into the bulk tank

(without testing)?

• Cost of test
• Loss from delivering infected milk to the dairy
• Reliability of test

Reliability of milk test 0.99 0.01

Infected = ”no”

0.01 0.99

Infected = ”yes” Test = negative Test = positive

Infected Test

0.9993 0.0007

Infected = ”no” Infected = ”yes”

What is the probability that a positive test is right?

Reliability of diagnostic tests

What is the probability that a Danish cow is suffering from “mad cow disease”? What are the sensitivity and specificity of the test for “mad cow disease”? Reasoning against the causal direction. Conclusion?

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In some situations it matters

• Honey, did you know that the prior likelihood of an event like this is extremely small?

The cow feeding example

Should we perform laboratory analyses of roughages fed to dairy cows? Depends on:

• The precision of the analysis
• The herd size
• The cost of the analyses
• The relative amount of roughages fed
• The uncertainty of the milk yield response to the nutritional

contents

For simplicity:

• Only energy content considered
• Only one roughage and one concentrate

The cow feeding example

The obvious action as a consequence of the test result is to adjust the proportion of roughages relative to concentrates in the ration.

The cow feeding examble

Test Action

Decision trees

A very common technique for evaluation of alternative decisions over time. In particular popular in the veterinary community. Example diseased calf:

• Treat: Yes/no
• Cost of treatment: 100 DKK
• Value of surviving calve: 1650 DKK
• Survival of animals:
• Treated: 0.88
• Untreated: 0.60

Decision tree for diseased calf

Value of decision “Yes”:

• 0.88 × 1650 + 0.12 × (-70) – 100 = 1343.60

Value of decision “No”:

• 0.60 × 1650 + 0.40 × (-70) = 962

The optimal decision is obviously to treat.

Treat Die Die

Y: -100 N: 0 N: 0.88 Y: 0.12

• 70

1650

• 70

1650

Untreated, dead Untreated, survived Treated, dead Treated, survived

N: 0.60 Y: 0.40

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The treatment problem as a decision graph

Refer to file “Treatment.xbn” The answers are (fortunately) the same as with the decision tree. Any decision graph can be modeled as a decision tree.

A tiny part of the cow feeding problem

Number of leaves: 3 × 4 × 5 × 5 = 300 (only 5 shown) That’s why we need decision graphs! Hidden assumptions (e.g. true value for silage).

Me HS Me Me Obs Obs

Obs

Obs Mix Mix Mix R R

R

R R Mix Mix