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## Making decisions using influence diagrams

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**Making decisions using influence diagrams**• An influence diagram can be converted into a decision tree that is a symmetric expansion of a conventional decision tree • Example Uncertain event Decision Consequence**Symmetric decision tree**Sunshine (p) Take Rain (1-p) Sunshine (p) Do not take Rain (1-p)**Procedure for making decisions**• Find chance nodes that can be replaced by expected value of payoff of uncertain event (e.g., weather) • Directly precede consequence node • Do not directly precede other node • Find decision that can be replaced by branch with optimum expected value (e.g., take or not take umbrella)**Example**Uncertain event Decision Consequence Uncertain event Decision Consequence Action Consequence Take 0.8 Do not take p Solution: Take umbrella if Pr(sunshine)<0.8, otherwise do not take umbrella**Example: Imperfect information**Original influence diagram Forecast Path Consequence Wait for forecast? Evacuate? Stay Evacuate Stay Evacuate**Challenge**• There is no chance node to reduce • Path of storm affects forecast • Solution: consider that the forecast is the fundamental uncertainty that influences the path of the storm. In terms of the influence diagram, reverse the arrow from the path to the forecast. • Need probabilities of forecasting hit (H’) and miss (M’) and conditional probabilities of hit and miss given that the forecast is hit, and miss. • Find these probabilities using Bayes’ rule.**Modified influence diagram**Forecast Path Consequence Wait for forecast? Evacuate? Stay Evacuate Stay Evacuate**Bayes’ rule**• Find probability of a hypothesis being true given the evidence, P(H/E) • Usually it is easier to estimate probability of getting evidence E given that the hypothesis is true, P(E/H)**Example**• One 4-sided die, one 20-sided die • Prior probabilities of picking the 4-sided and 20-sided = 0.5 • Evidence: One die was picked at random. We do not know which die was picked. We rolled the die once and got 3. • What is the updated probability of each die in the light of evidence?**Prior information**Evidence, E: 3 4-sided P=0.5 4-sided P=0.5 20-sided P=0.5 20-sided P=0.5 Example P(E/4-sided)=0.25 P(E/20-sided)=0.05**Example**• Hypothesis, H: 4-sided die was selected • P(E/H)=P(3/4-sided) = 0.25 • P(E/HC)=P(3/20-sided) = 0.05 • Posterior obtained from Bayes theorem.**ExamplePosterior Probabilities**4-sided P=0.833 20-sided P=0.167**Modified influence diagram**Forecast Path Consequence Wait for forecast? Evacuate? Stay Evacuate Stay Evacuate**Computing probabilities of the storm hitting and missing**Miami given that the forecast says so**Computing probabilities of the storm hitting and missing**Miami given that the forecast says it will miss it**First step in solving influence diagram**Forecast Consequence Wait for forecast? Evacuate? Stay Evacuate Stay Evacuate**Second step in solving influence diagram**Forecast Wait for forecast? Consequence Stay Evacuate**Third step in solving influence diagram**Wait for forecast? Consequence Stay Evacuate**Solution: Stay and wait for forecast. If forecast says**storm will hit Miami, evacuate. Otherwise stay. Value of additional information from forecast=0.874-0.8=0.074