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 conditional probabilities. They express C Child the influence of the values of the Parent 2 parents on the value of the child. Parent 1 Edges into a utility node correspond to a function depending on the values of = Decision variable, Decision node D D the parents. Child Edges into a decision node just means that the values of the parents are Parent 2 known when the decision is made. Parent 1 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 • More than one decision Choice 1 • 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 graph. Consider: • The decision to Test 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 • No influence on Flu Aspirin Slide 9 Slide 10 Causal influences How to model a test decision As long as we don’t {Not observed, normal, medium, high} include actions, the Obs. Flu Fever Tired Test two Bayesian fever networks are equivalent {No, yes} The causal direction If we include the action “Aspirin”, it would cure the flu in the Flu Fever Tired latter case! Flu Fever Tired In particular with {Normal, {No, yes} decision graphs: {No, yes} 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 Fever Test P( Obs. Fever | Fever, Test ) {Not observed, normal, medium, high} Not observed Normal Medium High Obs. 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? What is the benefit? Price Fever example: • Costs of test. • Improvement in “result” Will the result change your actions? Inconv Cost: • Are there any intervening options? Cost: • Can you trust the result of the test? Price of aspirin Inconvenience Obs. Test Aspirin fever Benefit Flu Fever Fever’ Tired Slide 15 Slide 16 What is the total benefit? Multi attribute utility: two attributes Consequences: • Less tired “Well-being” • Inconvenience of fever test • Price of aspirin Not comparable – cannot be added! We need a common unit for measurement ) u 21 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 All utilities must be expressed in the same unit (e.g. 2 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” What are the sensitivity and specificity of the test for Infected “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

The cow feeding example The cow feeding example Should we perform laboratory analyses of roughages fed to dairy The obvious action as a consequence of the test result is to adjust cows? the proportion of roughages relative to concentrates in the Depends on: ration. • 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 Slide 25 Slide 26 The cow feeding examble 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 • Cost of dead calf: 70 DKK • Survival of animals: • Treated: 0.88 • Untreated: 0.60 Action Test Slide 27 Slide 28 Decision tree for diseased calf The treatment problem as a decision graph N: 0.88 1650 Treated, survived Die Y: -100 -70 Treated, dead Y: 0.12 Treat 1650 Untreated, survived N: 0.60 N: 0 Die -70 Untreated, dead Y: 0.40 Refer to file “Treatment.xbn” Value of decision “Yes”: The answers are (fortunately) the same as with the decision tree. • 0.88 × 1650 + 0.12 × (-70) – 100 = 1343.60 Any decision graph can be modeled as a decision tree. 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 Mix R Obs Mix R Me Obs Mix R HS Me Obs Mix R Me Obs Mix R 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 6