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

Several names Decision graphs • Decision trees • Influence diagrams Decision graphs I • A certain kind of strictly symmetric decision trees Decisions and utilities • “No forgetting” assumption • LIMIDs – Limited Memory Influence Diagrams Anders Ringgaard Kristensen • Influence diagrams without the “no forgetting” assumption Very often the terms “Decision graphs” and “Influence diagrams” are used synonymously. Slide 1 Slide 2 Where are we? 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 Bayesian Decision Networks Graphs First processing: Monitoring & filtering Second processing: Decision making Some methods integrate the whole setup Slide 3 Slide 4 Parent 1 Parent 2 Notation, variables (= nodes) Numerical contents Edges into a chance node (yellow circle) correspond to a set of Random variable, Chance node C Child conditional probabilities. They express the influence of the values of the Parent 2 Parent 1 parents on the value of the child. Edges into a utility node correspond to = D Decision variable, Decision node D a function depending on the values of the parents. Child Edges into a decision node just means that the values of the parents are Parent 2 Parent 1 known when the decision is made. U U = Utility variable, Utility node They are called information edges. The decision may depend on the values of its parents. Child Slide 5 Slide 6 1

Baysian networks for decision analysis The TV show problem as a BN Problems Choice 1 • More than one decision • Many combinations to test • The same decision is not necessarily the best under all circumstances Refer do demo file ShowBN.xbn • Complex strategies • Cannot be entered as evidence Opened Choice 2 • Need for direct optimization Gain True Slide 7 Slide 8 Exercise Two kinds of decisions Use the notation on Slide 5 and the rules on Slide 6 to Test decisions change the TV show problem on Slide 7 to a decision Test • The decision to graph. Consider: observe the value of • Which nodes are actually decision nodes (from the a variable. player’s point of view)? Actions • Should we add any information edges? • Influences the value Flu Fever Tired of at least one variable. • In this case: • Fever • Tired Aspirin • No influence on Flu Slide 9 Slide 10 Causal influences How to model a test decision {Not observed, normal, medium, high} As long as we don’t Obs. include actions, the Flu Fever Tired Test fever two Bayesian networks are {No, yes} equivalent The causal direction If we include the action “Aspirin”, it would Flu Fever Tired cure the flu in the Flu Fever Tired latter case! In particular with {Normal, {No, yes} {No, yes} decision graphs: medium, The reasoning direction • Be careful with high} causality! Include a “mirror variable” for the observed node: • The “mirror” has the same states AND a “not observed” state. Slide 11 Slide 12 2

P( Obs. Fever | Fever, Test ) How to model the action {Not observed, normal, medium, high} Fever Test P( Obs. Fever | Fever, Test ) Obs. Not observed Normal Medium High Test Aspirin {No, yes} fever Normal Yes 0 1 0 0 {No, yes} Medium Yes 0 0 1 0 High Yes 0 0 0 1 Flu Fever Fever’ Tired Normal No 1 0 0 0 Medium No 1 0 0 0 {Normal, {Normal, {No, yes} {No, yes} medium, medium, High No 1 0 0 0 high} high} 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. Slide 13 Slide 14 Does it pay to perform tests? What is the benefit? Price What is the benefit? Fever example: • Costs of test. • Improvement in “result” Will the result change your actions? Inconv Cost: Cost: • Are there any intervening options? Price of aspirin Inconvenience • Can you trust the result of the test? Obs. Test Aspirin fever Benefit Flu Fever Fever’ Tired Slide 15 Slide 16 What is the total benefit? Multi attribute utility: two attributes Consequences: “Well-being” • Less tired • Inconvenience of fever test • Price of aspirin Not comparable – cannot be added! ) u 21 We need a common unit for measurement The utility concept – a well known theory High ) u 22 Medium Low 0 ) u ) u 1 1 Monetary gain Slide 17 Slide 18 3

Utility concept in decision graphs Constant marginal rate of substitution Decision graphs don’t support varying marginal rates of Well being substitution between attributes. The values of all utility nodes of the graph are just summed and maximized. ) u 2 All utilities must be expressed in the same unit (e.g. money). No time preference allowed. ) u 2 Low Medium High 0 ) u ) u 1 Monetary gain 1 Slide 19 Slide 20 Will the result change your actions? Will the result change your actions? The fever example: The simple milk test example (example from Tuesday): • The aspirin action is the obvious and natural consequence of • The obvious action is not to pour the milk from a positive the test. test into the bulk tank. • But, what about just taking an aspirin (without testing)? • We would never do that unless we think the milk is infected. • Inconvenience of test • But, what about just pouring the milk into the bulk tank (without testing)? • Cost of aspirin • Cost of test • “You are not allowed to do that” – if true, a utility node is missing! • Loss from delivering infected milk to the dairy • Reliability of test • Reliability of test Slide 21 Slide 22 Reliability of milk test Reliability of diagnostic tests What is the probability that a Danish cow is suffering from “mad cow disease”? Infected = ”yes” Infected = ”no” Infected What are the sensitivity and specificity of the test for “mad cow disease”? 0.0007 0.9993 Reasoning against the causal direction. Conclusion? Test = positive Test = negative Test Infected = ”yes” 0.99 0.01 Infected = ”no” 0.01 0.99 What is the probability that a positive test is right? Slide 23 Slide 24 4

In some situations it matters 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: Morgenthaler in Politiken: • Only energy content considered • Honey, did you know that the prior likelihood of an event like this is extremely small? • Only one roughage and one concentrate Slide 25 Slide 26 The cow feeding examble 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. Action Test Slide 27 Slide 28 Decision tree for diseased calf Decision trees A very common technique for evaluation of alternative decisions over N: 0.88 1650 Treated, survived time. Y: -100 Die In particular popular in the veterinary community. -70 Treated, dead Y: 0.12 Example diseased calf: Treat • Treat: Yes/no N: 0.60 1650 Untreated, survived • Cost of treatment: 100 DKK N: 0 Die • Value of surviving calve: 1650 DKK -70 Untreated, dead Y: 0.40 • Survival of animals: • Treated: 0.88 Value of decision “Yes”: • Untreated: 0.60 • 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. Slide 29 Slide 30 5

A tiny part of the cow feeding problem The treatment problem as a decision graph Mix R Obs Mix R Me Obs Mix R HS Me Obs Mix R Refer to file “Treatment.xbn” Me Obs Mix R The answers are (fortunately) the same as with the decision tree. Any decision graph can be modeled as a decision tree. 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). Slide 31 Slide 32 6