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1 Chap. 4 Decision Graphs Statistical Genetics Forum Bayesian Networks and Decision Graphs Finn V. Jensen Presented by Ken Chen Genome Sequencing Center

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2 Flu Fever Sleepy T A Using probabilities provided by network to support decision-making Test decisions Look for more evidences Action decisions

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3 OH0OH1 FCSC OH2 BH MH OH0OH1 FCSC OH2 BHMH D U One Action: Example: CallFold Poker Game:

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4 One action in general D Goal: find D=d that maximize EU(D|e)

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5 Action GA DSS U GA U DSS 4.2 Utilities: Example: Management of effort Decision: Gd: keep pace in GA, follow DSS superficially SB: slow down in both courses Dg: keep pace in DSS, follow GA superficially Game 1: maximize the sum of the exp marks General: maximize the sum of the exp utilities

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6 4.3 Value of information A H U T

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7 Nonutility value functions When there is no proper model for actions and utilities, the reason for test is to decrease the uncertainty of the hypothesis

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8 Test 1 Test 2 Action T2T2 Inf 99.94 -0.06 97.74 Inf 99.74 -0.26 T1T1 yes pos neg yes pour pos neg discard pour clean infectedclean infected Nonmyopic data request

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9 Decision Tree Nonleaf nodes are decision nodes or chance nodes, and the leaves are utility nodes Complete: For a chance node there must be a link for each possible state, and from a decision node there must be a link for each possible decision option D action 1 action 2 action n … X P(X=x 1 |o) P(X=x 2 |o) P(X=x n |o) … U

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10 A car start problem Possible Fault: –Spark Plug (SP), prob=0.3 –Ignition System (IS), prob=0.2, –Others, prob=0.5 Actions: –SP, fixes SP, 4 min –IS, fixes IS with prob=0.5, 2 min –T, test OK iff. IS is OK, 0.5 min –RS, fixes everything, 15 min Goal: –Have car fixed asap

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11 15 D D D D D 10.514.525.5 D D D D D D 27.5 12.5 14.5 D SP 28 26 OK !OK RS SP OK !OK 0.38 0.62 0.8 0.2 IS !OK OK RS 0.5 RS OK !OK 0.7 0.3 RS OK !OK 0.1 0.9 P(SP fix|T=OK) =P(SP|T=OK) =P(SP| !IS ) =P(SP)/(P(SP)+P(others)) =0.3/0.8=0.38 P(IS fix)=P(IS)P(fix|IS)=0.2*0.5=0.1 Fault TFault-I IS … … … … RS T IS

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12 15 D D D D D 10.514.525.5 D D D D D D 27.5 12.5 14.5 D RS T SP IS 28 26 OK !OK RS SP OK !OK 0.38 0.62 0.8 0.2 IS !OK OK RS 0.5 RS OK !OK 0.7 0.3 RS OK !OK 0.1 0.9 16.96 16.27 15.43 … … … … Solving Decision Trees

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13 Coalesced decision trees Grow exponentially with the number of decisions and chance variables When decision tree contains identical subtrees they can be collapsed.

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14 4.5 Decision-Theoretic Troubleshooting A fault causing a device to malfunction is identified and eliminated through a sequence of troubleshooting steps. A troubleshooting problem can be represented and solved through a decision tree (actions and questions) As decision trees have a risk of becoming intractably large, we look for ways of pruning the decision tree.

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15 Action sequences A i =yes, A i =no Cost of action A i, C i ( ), evidences Action seq: s= repeatly performing the next action until problem gets fixed or the last action has been performed Expected cost of repair (ECR)

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16 Local optimality of the optimal sequence (Dynamic Programming) Consider two neighboring actions A i and A i+1 Pruned tree has eight non-RS links, compared to 32 in a coalesced DT for the same problem

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17 The greedy approach Always choose the action with the highest efficiency Not necessarily optimal! Proposition 4.2: Conditions under which the greedy approach is optimal: –n faults F 1 …F n, and n actions: A 1 … A n –Exactly one of the faults is present –Each action has a specific probability of repair: p i =P(A i =yes|F i ), P(A i =yes|F j )=0 if i≠j –The cost C i of an action does not depend on the performance of previous actions Theorem 4.2: for action sequence s fulfilling the conditions in Proposition 4.2. Assume s is ordered according to decreasing initial efficiencies. Then s is an optimal action sequence and F1 F2 F3 F4 A1 A2 A3

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18 Influence Diagram A compact representation of decision tree Now seen more as a decision tool extending Bayesian networks Syntax: –There is a directed path comprising all decision nodes –The utility nodes have no children –The decision nodes and the chance nodes have a finite set of mutually exclusive states –The utility nodes have no states –To each chance node A is attached a conditional probability table P(A|pa(A)) –The each utility node U is attached a real-valued function over pa(U)

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19 OH0OH1 OFCOSC OH BH MH D U MH0MH1 MFCMSC BN Influence Diagram OH0OH1 OFCOSC OH BH MH D U MH0MH1 MFCMSC No-forgetting: The decision maker remembers the past observations and decisions

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20 Solution to influence diagrams Similar to decision-tree More efficiently by exploiting the structure of of the influence diagram (Chapter 7)

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21 Information blocking V1V1 T1T1 FV 1 U1U1 V2V2 T2T2 FV 2 U2U2 V3V3 T3T3 FV 3 U3U3 V4V4 T4T4 FV 4 U4U4 V5V5 T5T5 FV 5 U5U5 FV 5 has 10 9 elements V1V1 T1T1 FV 1 U1U1 V2V2 T2T2 FV 2 U2U2 V3V3 T3T3 FV 3 U3U3 V4V4 T4T4 FV 4 U4U4 V5V5 T5T5 FV 5 U5U5 Introduce variables/links which,when observed, d-separate most of the past from The present decision Fishing Vol

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