# Decision Trees and Networks

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1 Lecture 21: Uncertainty 6 Today s Lecture Victor R. Lesser CMPSCI 683 Fall 2010 Decision Trees and Networks Decision Trees A decision tree is an explicit representation of all the possible scenarios from a given state. Each path corresponds to decisions made by the agent, actions taken, possible observations, state changes, and a final outcome node. Similar to a game played against nature - Decision node - Chance node Display! Software Example 1: Software Development make! buy reuse major changes P=0.6 simple P=0.3 difficult P=0.7 minor changes P=0.4 simple P=0.2 complex P=0.8 minor changes P=0.7 major changes P=0.3 \$380K \$450K \$275K \$310K \$490K \$210K \$400K EU(make) = 0.3 \$380K \$450K = \$429K EU(reuse) = 0.4 \$275K [0.2 \$310K \$490K] = \$382.4K EU(buy) = 0.7 \$210K \$400K = \$267K ; best choice

4 Example 2: Buying a car cont. Decision Networks/Influence Diagrams fail pass T 2 T 1 fail T 0 pass Decision networks or influence diagrams are an extension of belief networks that allow for reasoning about actions and utility. C 1 C The network represents information about the agent s current state, its possible actions, the possible outcome of those actions, and their utility Nodes in a Decision Network Example 3: Taking an Umbrella Chance nodes (ovals) have CPTs (conditional probability tables) that depend on the states of the parent nodes (chance or decision). Decision nodes (squares) represent options available to the decision maker. Utility nodes (Diamonds) or value nodes represent the overall utility based on the states of the parent nodes. Rain (hidden variable) Does WeatherReport Indicate Rain? Bring Umbrella? Utility Parameters: P(Rain), P(WeatherReport Rain), P(WeatherReport Rain), Utility(Rain,Umbrella)

5 Taking an Umbrella as Decision Tree Knowledge in an Influence Diagram Rain Causal knowledge about how events influence each other in the domain Weather Report Yes (take umbrella) Rain No (don t take umbrella) Not Rain Yes No Knowledge about what action sequences are feasible in any given set of circumstances Lays out possible temporal ordering of decisions 1 Umbrella Not Umbrella Umbrella Not Umbrella Normative (Utility) knowledge about how desirable the consequences are Example 2 as an Influence Diagram Decision Trees vs Influence Diagrams C1 T1 Cost of test T is decision whether to do a Test or not and which one D is the decision of which car to buy T D r(c1,d) T2 r(t) C2 r(c2,d) fail T 2 pass C 1 T 1 T 0 fail pass C Decision trees are not convenient for representing domain knowledge Requires tremendous amount of storage Multiple decisions nodes -- expands tree Duplication of knowledge along different paths Joint Probability Distribution vs Bayes Net Generate decision tree on the fly from more economical forms of knowledge Depth-first expansion of tree for computing optimal decision

6 Topology of decision networks 1. The directed graph has no cycles. 2. The utility nodes have no children. 3. There is a directed path that contains all of the decision nodes. 4. A CPT is attached to each chance node specifying P(A parents(a)). 5. A real valued function over parents(u) is attached to each utility node. Semantics Links into decision nodes are called information links, and they indicate that the state of the parent is known prior to the decision. The directed path that goes through all the decision nodes defines a temporal sequence of decisions. It also partitions the chance variables into sets: I 0 is the vars observed before any decision is made, I 1 is the vars observed after the first and before the second decision, etc. I n is the set of unobserved vars. The no-forgetting assumption is that the decision maker remembers all past observations and decisions. -- Non Markov Assumption Example 4: Airport Siting Problem Evaluating Decision Networks Airport Site 1. Set the evidence variables for the current state. Air Traffic! Litigation Deaths Noise! Utility (deaths,noise,cost) U 2. For each possible value of the decision node(s): (a) Set the decision node to that value. (b) Calculate the posterior probabilities for the parent nodes of the utility node. (c) Calculate the expected utility for the action. 3. Return the action/decision with the highest utility. Construction! Cost P(cost=high airportsite=darien, airtraffic=low, litigation=high, construction=high)

7 Imperfect Information Example 5: Mildew Two months before the harvest of a wheat field, the farmer observes the state Q of the crop, and he observes whether it has been attacked by mildew, M. If there is an attack, he will decide on a treatment with fungicides. Mildew decision model Maximize ( value of crop - cost of action ) Q H U Value of crop There are five variables: - Q: fair (f), not too bad (n), average (a), good (g) - M: no (no), little (l), moderate (m), severe (s) - H: state of Q plus M: rotten (r),bad (b), poor (p) OQ M M* State of Mildew after Treatment - OQ: observation of Q; imperfect information on Q - OM: observation of M; imperfect information on M OM A V Cost of action One action in general Multiple decisions -- Policy Generation A single decision node D may have links to some chance nodes. A set of utility functions U 1,,U n over domains X 1,,X n. Goal: find the decision d that maximizes EU(D=d e): C1 T1 T D T2 C2 How to solve such problems using a standard Bayesian network package? V Need a more complex evaluation technique since generating a policy (sequence of decisions)

9 Barren Node Removal becomes becomes Notation for Shachter s algorithm For chance nodes: S(i) = direct successors = children C(i) = conditional predecessors = parents For decision nodes I(i) = information predecessors = parents Chance Node Removal i C(i) \ C(v) C(i) C(v) C(v)\C(i)\{i} nodes connected to i but not to v becomes v Node i directly linked to utility node v nodes connected to v but not to i and not i v marginalization C(i) \ C(v) C(i) C(v) C(v)\C(i)\{i}

10 Decision node removal Arc reversal I(i) \ C(v) i I(i) C(v) v becomes maximization v Given an influence diagram containing an arc from i to j, but no other directed path from i to j, it is possible to to transform the diagram to one with an arc from j to i. (If j is deterministic, then it becomes probabilistic.) I(i) C(v) Arc Reversal i j Bayes rule C(i) \ C(j) C(i) C(j) C(j)\C(i)\{i} becomes i j C(i) \ C(j) C(i) C(j) C(j)\C(i)\{i} Pa=Parents Pa(A)\Pa(B) parents of A who are not parents of B

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