Decision Making Under Uncertainty CMSC 671 – Fall 2010 R&N, Chapters , , material from Lise Getoor, Jean-Claude Latombe, and Daphne Koller 1

Decision Making Under Uncertainty Many environments have multiple possible outcomes Some of these outcomes may be good; others may be bad Some may be very likely; others unlikely What’s a poor agent to do?? 2

Non-Deterministic vs. Probabilistic Uncertainty ? bac {a,b,c}  decision that is best for worst case ? bac {a(p a ), b(p b ), c(p c )}  decision that maximizes expected utility value Non-deterministic modelProbabilistic model ~ Adversarial search 3

Expected Utility Random variable X with n values x 1,…,x n and distribution (p 1,…,p n ) E.g.: X is the state reached after doing an action A under uncertainty Function U of X E.g., U is the utility of a state The expected utility of A is EU[A] =  i=1,…,n p(x i |A)U(x i ) 4

s0s0 s3s3 s2s2 s1s1 A U(A1, S0) = 100 x x x 0.1 = = 62 One State/One Action Example 5

s0s0 s3s3 s2s2 s1s1 A A2 s4s U (A1, S0) = 62 U (A2, S0) = 74 U (S0) = max a {U(a,S0)} = 74 One State/Two Actions Example 6

s0s0 s3s3 s2s2 s1s1 A A2 s4s U (A1, S0) = 62 – 5 = 57 U (A2, S0) = 74 – 25 = 49 U (S0) = max a {U(a, S0)} = Introducing Action Costs 7

MEU Principle A rational agent should choose the action that maximizes agent’s expected utility This is the basis of the field of decision theory The MEU principle provides a normative criterion for rational choice of action 8

Not quite… Must have a complete model of: Actions Utilities States Even if you have a complete model, decision making is computationally intractable In fact, a truly rational agent takes into account the utility of reasoning as well (bounded rationality) Nevertheless, great progress has been made in this area recently, and we are able to solve much more complex decision-theoretic problems than ever before 9

Axioms of Utility Theory Orderability (A>B)  (A<B)  (A~B) Transitivity (A>B)  (B>C)  (A>C) Continuity A>B>C   p [p,A; 1-p,C] ~ B Substitutability A~B  [p,A; 1-p,C]~[p,B; 1-p,C] Monotonicity A>B  (p≥q  [p,A; 1-p,B] >~ [q,A; 1-q,B]) Decomposability [p,A; 1-p, [q,B; 1-q, C]] ~ [p,A; (1-p)q, B; (1-p)(1-q), C] 10

Money Versus Utility Money <> Utility More money is better, but not always in a linear relationship to the amount of money Expected Monetary Value Risk-averse: U(L) < U(S EMV(L) ) Risk-seeking: U(L) > U(S EMV(L) ) Risk-neutral: U(L) = U(S EMV(L) ) 11

Value Function Provides a ranking of alternatives, but not a meaningful metric scale Also known as an “ordinal utility function” Sometimes, only relative judgments (value functions) are necessary At other times, absolute judgments (utility functions) are required 12

Multiattribute Utility Theory A given state may have multiple utilities...because of multiple evaluation criteria...because of multiple agents (interested parties) with different utility functions We will talk about this more later in the semester, when we discuss multi-agent systems and game theory 13

Decision Networks Extend BNs to handle actions and utilities Also called influence diagrams Use BN inference methods to solve Perform Value of Information calculations 14

Decision Networks cont. Chance nodes: random variables, as in BNs Decision nodes: actions that a decision maker can take Utility/value nodes: the utility of an outcome state 15

R&N example 16

Umbrella Network weather forecast umbrella happiness take/don’t take f w p(f|w) sunny rain 0.3 rainy rain 0.7 sunny no rain 0.8 rainy no rain 0.2 P(rain) = 0.4 U(have,rain) = -25 U(have,~rain) = 0 U(~have, rain) = -100 U(~have, ~rain) = 100 have umbrella P(have|take) = 1.0 P(~have|~take)=1.0 17

Evaluating Decision Networks Set the evidence variables for current state For each possible value of the decision node: Set decision node to that value Calculate the posterior probability of the parent nodes of the utility node, using BN inference Calculate the resulting utility for each action Return the action with the highest utility 18

Decision Making: Umbrella Network weather forecast umbrella happiness take/don’t take f w p(f|w) sunny rain 0.3 rainy rain 0.7 sunny no rain 0.8 rainy no rain 0.2 P(rain) = 0.4 U(have,rain) = -25 U(have,~rain) = 0 U(~have, rain) = -100 U(~have, ~rain) = 100 have umbrella P(have|take) = 1.0 P(~have|~take)=1.0 Should I take my umbrella?? 19

Value of Information (VOI) Suppose an agent’s current knowledge is E. The value of the current best action  is: The value of the new best action (after new evidence E’ is obtained): The value of information for E’ is therefore: 20

Value of Information: Umbrella Network weather forecast umbrella happiness take/don’t take f w p(f|w) sunny rain 0.3 rainy rain 0.7 sunny no rain 0.8 rainy no rain 0.2 P(rain) = 0.4 U(have,rain) = -25 U(have,~rain) = 0 U(~have, rain) = -100 U(~have, ~rain) = 100 have umbrella P(have|take) = 1.0 P(~have|~take)=1.0 What is the value of knowing the weather forecast? 21

Sequential Decision Making Finite Horizon Infinite Horizon 22

Simple Robot Navigation Problem In each state, the possible actions are U, D, R, and L 23

Probabilistic Transition Model In each state, the possible actions are U, D, R, and L The effect of U is as follows (transition model): With probability 0.8, the robot moves up one square (if the robot is already in the top row, then it does not move) 24

Probabilistic Transition Model In each state, the possible actions are U, D, R, and L The effect of U is as follows (transition model): With probability 0.8 the robot moves up one square (if the robot is already in the top row, then it does not move) With probability 0.1 the robot moves right one square (if the robot is already in the rightmost row, then it does not move) 25

Probabilistic Transition Model In each state, the possible actions are U, D, R, and L The effect of U is as follows (transition model): With probability 0.8 the robot moves up one square (if the robot is already in the top row, then it does not move) With probability 0.1 the robot moves right one square (if the robot is already in the rightmost row, then it does not move) With probability 0.1 the robot moves left one square (if the robot is already in the leftmost row, then it does not move) 26

Markov Property The transition properties depend only on the current state, not on the previous history (how that state was reached) 27

Sequence of Actions Planned sequence of actions: (U, R) [3,2] 28

Sequence of Actions Planned sequence of actions: (U, R) U is executed [3,2] [4,2][3,3][3,2] 29

Histories Planned sequence of actions: (U, R) U has been executed R is executed There are 9 possible sequences of states – called histories – and 6 possible final states for the robot! [3,2] [4,2][3,3][3,2] [3,3][3,2][4,1][4,2][4,3][3,1] 30

Probability of Reaching the Goal P([4,3] | (U,R).[3,2]) = P([4,3] | R.[3,3]) x P([3,3] | U.[3,2]) + P([4,3] | R.[4,2]) x P([4,2] | U.[3,2]) Note importance of Markov property in this derivation P([3,3] | U.[3,2]) = 0.8 P([4,2] | U.[3,2]) = 0.1 P([4,3] | R.[3,3]) = 0.8 P([4,3] | R.[4,2]) = 0.1 P([4,3] | (U,R).[3,2]) =

Utility Function [4,3] provides power supply [4,2] is a sand area from which the robot cannot escape

Utility Function [4,3] provides power supply [4,2] is a sand area from which the robot cannot escape The robot needs to recharge its batteries

Utility Function [4,3] provides power supply [4,2] is a sand area from which the robot cannot escape The robot needs to recharge its batteries [4,3] or [4,2] are terminal states

Utility of a History [4,3] provides power supply [4,2] is a sand area from which the robot cannot escape The robot needs to recharge its batteries [4,3] or [4,2] are terminal states The utility of a history is defined by the utility of the last state (+1 or –1) minus n/25, where n is the number of moves

Utility of an Action Sequence +1 Consider the action sequence (U,R) from [3,2]

Utility of an Action Sequence +1 Consider the action sequence (U,R) from [3,2] A run produces one of 7 possible histories, each with some probability [3,2] [4,2][3,3][3,2] [3,3][3,2][4,1][4,2][4,3][3,1] 37

Utility of an Action Sequence +1 Consider the action sequence (U,R) from [3,2] A run produces one among 7 possible histories, each with some probability The utility of the sequence is the expected utility of the histories: U =  h U h P(h) [3,2] [4,2][3,3][3,2] [3,3][3,2][4,1][4,2][4,3][3,1] 38

Optimal Action Sequence +1 Consider the action sequence (U,R) from [3,2] A run produces one among 7 possible histories, each with some probability The utility of the sequence is the expected utility of the histories The optimal sequence is the one with maximal utility [3,2] [4,2][3,3][3,2] [3,3][3,2][4,1][4,2][4,3][3,1] 39

Optimal Action Sequence +1 Consider the action sequence (U,R) from [3,2] A run produces one among 7 possible histories, each with some probability The utility of the sequence is the expected utility of the histories The optimal sequence is the one with maximal utility But is the optimal action sequence what we want to compute? [3,2] [4,2][3,3][3,2] [3,3][3,2][4,1][4,2][4,3][3,1] only if the sequence is executed blindly! 40

Accessible or observable state Repeat:  s  sensed state  If s is terminal then exit  a  choose action (given s)  Perform a Reactive Agent Algorithm 41

Policy (Reactive/Closed-Loop Strategy) A policy  is a complete mapping from states to actions

Repeat:  s  sensed state  If s is terminal then exit  a   (s)  Perform a Reactive Agent Algorithm 43

Optimal Policy +1 A policy  is a complete mapping from states to actions The optimal policy  * is the one that always yields a history (ending at a terminal state) with maximal expected utility Makes sense because of Markov property Note that [3,2] is a “dangerous” state that the optimal policy tries to avoid 44

Optimal Policy +1 A policy  is a complete mapping from states to actions The optimal policy  * is the one that always yields a history with maximal expected utility This problem is called a Markov Decision Problem (MDP) How to compute  *? 45

Additive Utility History H = (s 0,s 1,…,s n ) The utility of H is additive iff: U (s 0,s 1,…,s n ) = R (0) + U (s 1,…,s n ) =  R (i) Reward 46

Additive Utility History H = (s 0,s 1,…,s n ) The utility of H is additive iff: U (s 0,s 1,…,s n ) = R (0) + U (s 1,…,s n ) =  R (i) Robot navigation example: R (n) = +1 if s n = [4,3] R (n) = -1 if s n = [4,2] R (i) = -1/25 if i = 0, …, n-1 47

Principle of Max Expected Utility History H = (s 0,s 1,…,s n ) Utility of H: U (s 0,s 1,…,s n ) =  R (i)  First-step analysis  U (i) = R (i) + max a  k P (k | a.i) U (k)  *(i) = arg max a  k P (k | a.i) U (k) +1 48

Defining State Utility Problem: When making a decision, we only know the reward so far, and the possible actions We’ve defined utility retroactively (i.e., the utility of a history is obvious once we finish it) What is the utility of a particular state in the middle of decision making? Need to compute expected utility of possible future histories 49

Value Iteration Initialize the utility of each non-terminal state s i to U 0 (i) = 0 For t = 0, 1, 2, …, do: U t+1 (i)  R (i) + max a  k P (k | a.i) U t (k)

Value Iteration Initialize the utility of each non-terminal state s i to U 0 (i) = 0 For t = 0, 1, 2, …, do: U t+1 (i)  R (i) + max a  k P (k | a.i) U t (k) U t ([3,1]) t Note the importance of terminal states and connectivity of the state-transition graph 51

Policy Iteration Pick a policy  at random 52

Policy Iteration Pick a policy  at random Repeat: Compute the utility of each state for  U t+1 (i)  R (i) +  k P (k |  (i).i) U t (k) 53

Policy Iteration Pick a policy  at random Repeat: Compute the utility of each state for  U t+1 (i)  R (i) +  k P (k |  (i).i) U t (k) Compute the policy  ’ given these utilities  ’(i) = arg max a  k P (k | a.i) U (k) 54

Policy Iteration Pick a policy  at random Repeat: Compute the utility of each state for  U t+1 (i)  R (i) +  k P (k |  (i).i) U t (k) Compute the policy  ’ given these utilities  ’(i) = arg max a  k P(k | a.i) U (k) If  ’ =  then return  Or solve the set of linear equations: U (i) = R (i) +  k P (k |  (i).i) U (k) (often a sparse system) 55

Infinite Horizon In many problems, e.g., the robot navigation example, histories are potentially unbounded and the same state can be reached many times One trick: Use discounting to make an infinite horizon problem mathematically tractable What if the robot lives forever? 56