CS计算机代考程序代写 chain Agent-based Systems

Agent-based Systems
Paolo Turrini
™ www.dcs.warwick.ac.uk/~pturrini R p.turrini@warwick.ac.uk

Risks and Decisions Knowing what to expect
Paolo Turrini Risk and Decisions Agent-based Systems

The plan for today
Probabilities: basics
Bayes’ rule and conditional independence Back to the Wumpus World
Paolo Turrini Risk and Decisions Agent-based Systems

The book
Stuart Russell and Peter Norvig
Artificial Intelligence: a modern approach
Chapters 13-14
Paolo Turrini Risk and Decisions Agent-based Systems

Holiday
You are back from a holiday on an exotic island, and your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it).
How worried should you be?
Paolo Turrini Risk and Decisions Agent-based Systems

Probability basics
Begin with a set Ω—the sample space e.g., 6 possible rolls of a dice.
w ∈ Ω is a sample point/possible world/atomic event
Paolo Turrini Risk and Decisions Agent-based Systems

Probability basics
A probability space or probability model is a sample space Ω with an assignment P(w) for every w ∈ Ω s.t.
0 􏰤 P(w) 􏰤 1
ΣwP(w) = 1
e.g., P(1)=P(2)=P(3)=P(4)=P(5)=P(6)=1/6.
Paolo Turrini Risk and Decisions Agent-based Systems

Events
An event A is any subset of Ω P(A) = Σ{w∈A}P(w)
E.g., P(dice roll < 4) = P(1) + P(2) + P(3) = 1/6 + 1/6 + 1/6 = 1/2 Paolo Turrini Risk and Decisions Agent-based Systems Random variables A random variable is a function from sample points to some range, e.g., R, [0, 1],{true, false} . . . e.g., Odd(1)=true. P induces a probability distribution for any random variable X: P(X =xi) = Σ{w:X(w)=xi}P(w) e.g., P(Odd = true) = P(1) + P(3) + P(5) = 1/6 + 1/6 + 1/6 = 1/2 Paolo Turrini Risk and Decisions Agent-based Systems Conditional probability Definition of conditional probability: P(a|b) = P(a ∧ b) if P(b) ̸= 0 P(b) Product rule gives an alternative formulation: P(a ∧ b) = P(a|b)P(b) = P(b|a)P(a) but then... Paolo Turrini Risk and Decisions Agent-based Systems Bayes’ rule Theorem (Bayes’ Rule) P(a|b) = P(b|a)P(a) P(b) Paolo Turrini Risk and Decisions Agent-based Systems Bayes’ rule Useful for assessing causal probability from diagnostic probability: P(Cause|Effect) = P(Effect|Cause)P(Cause) P (Effect ) E.g., let c be cold, s be sore throat: P(c|s) = P(s|c)P(c) = 0.9 × 0.001 = 0.18 P (s ) 0.005 Paolo Turrini Risk and Decisions Agent-based Systems Bayes’ rule P(c|s) = P(s|c)P(c) = 0.9 × 0.001 = 0.18 P (s ) 0.005 We might not know the prior probability of the evidence P(S) In this case... we compute the posterior probability for each value of the query variable (c,¬c) and then normalise by a normalisation constant α P(C|s) = α⟨P(s|c)P(c),P(s|¬c)P(¬c)⟩ Paolo Turrini Risk and Decisions Agent-based Systems Bayes’ rule with random variables Theorem (Bayes’ rule with random variables) P(X|Y) = αP(Y|X)P(X) Paolo Turrini Risk and Decisions Agent-based Systems Holiday solved You are back from a holiday on an exotic island, and your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). Paolo Turrini Risk and Decisions Agent-based Systems Holiday solved You are back from a holiday on an exotic island, and your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let d be disease, p be that you scored positive at the test: Paolo Turrini Risk and Decisions Agent-based Systems Holiday solved You are back from a holiday on an exotic island, and your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let d be disease, p be that you scored positive at the test: P(d|p) Paolo Turrini Risk and Decisions Agent-based Systems Holiday solved You are back from a holiday on an exotic island, and your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let d be disease, p be that you scored positive at the test: P(d|p) = P(p|d)P(d) P(p|d)P(d)+P(p|¬d)P(¬d) Paolo Turrini Risk and Decisions Agent-based Systems Holiday solved You are back from a holiday on an exotic island, and your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let d be disease, p be that you scored positive at the test: P(d|p) = P(p|d)P(d) = 0.99×0.0001 P(p|d)P(d)+P(p|¬d)P(¬d) 0.99×0.0001+0.01×0.9999 Paolo Turrini Risk and Decisions Agent-based Systems Holiday solved You are back from a holiday on an exotic island, and your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let d be disease, p be that you scored positive at the test: P(d|p) = P(p|d)P(d) = 0.99×0.0001 = 0.0098 P(p|d)P(d)+P(p|¬d)P(¬d) 0.99×0.0001+0.01×0.9999 Paolo Turrini Risk and Decisions Agent-based Systems Holiday solved You are back from a holiday on an exotic island, and your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let d be disease, p be that you scored positive at the test: P(d|p) = P(p|d)P(d) = 0.99×0.0001 = 0.0098 P(p|d)P(d)+P(p|¬d)P(¬d) 0.99×0.0001+0.01×0.9999 Notice:the posterior probability of disease is still very small! Paolo Turrini Risk and Decisions Agent-based Systems Independence A and B are independent iff P(A|B) = P(A) or P(B|A) = P(B) or = P(cavity) = P(cavity|Weather) = P(cavity|CR7dives) P(A, B) = P(A)P(B) Paolo Turrini Risk and Decisions Agent-based Systems Combining evidence Start with the joint distribution: P(Cavity |toothache ∧ catch) = Paolo Turrini Risk and Decisions Agent-based Systems Combining evidence Start with the joint distribution: P(Cavity |toothache ∧ catch) = α ⟨0.108, 0.016⟩ Paolo Turrini Risk and Decisions Agent-based Systems Combining evidence Start with the joint distribution: P(Cavity |toothache ∧ catch) = α ⟨0.108, 0.016⟩ = Paolo Turrini Risk and Decisions Agent-based Systems Combining evidence Start with the joint distribution: P(Cavity |toothache ∧ catch) = α ⟨0.108, 0.016⟩ = ⟨0.871, 0.129⟩ Paolo Turrini Risk and Decisions Agent-based Systems Combining evidence Start with the joint distribution: It doesn’t scale up to a large number of variables Can we simplify? Paolo Turrini Risk and Decisions Agent-based Systems Combining evidence We can’t use absolute independence: Toothache and Catch are not independent: If the probe catches in the tooth then it is likely the tooth has a cavity, which means that toothache is likely too. But they are independent given the presence or the absence of cavity! Toothache depends on the state of the nerves in the tooth, catch depends on the dentist’s skills, to which toothache is irrelevant Paolo Turrini Risk and Decisions Agent-based Systems Conditional independence 1 P(catch|toothache,cavity) = P(catch|cavity), the same independence holds if I haven’t got a cavity: 2 P(catch|toothache, ¬cavity) = P(catch|¬cavity) Catch is conditionally independent of Toothache given Cavity: Paolo Turrini Risk and Decisions Agent-based Systems Conditional independence contd. Write out full joint distribution using chain rule: P(Toothache,Catch,Cavity) Paolo Turrini Risk and Decisions Agent-based Systems Conditional independence contd. Write out full joint distribution using chain rule: P(Toothache,Catch,Cavity) = P(Toothache|Catch,Cavity)P(Catch,Cavity) Paolo Turrini Risk and Decisions Agent-based Systems Conditional independence contd. Write out full joint distribution using chain rule: P(Toothache,Catch,Cavity) = P(Toothache|Catch,Cavity)P(Catch,Cavity) = P(Toothache|Catch,Cavity)P(Catch|Cavity)P(Cavity) Paolo Turrini Risk and Decisions Agent-based Systems Conditional independence contd. Write out full joint distribution using chain rule: P(Toothache,Catch,Cavity) = P(Toothache|Catch,Cavity)P(Catch,Cavity) = P(Toothache|Catch,Cavity)P(Catch|Cavity)P(Cavity) = P(Toothache|Cavity)P(Catch|Cavity)P(Cavity) Paolo Turrini Risk and Decisions Agent-based Systems Conditional independence contd. Write out full joint distribution using chain rule: P(Toothache,Catch,Cavity) = P(Toothache|Catch,Cavity)P(Catch,Cavity) = P(Toothache|Catch,Cavity)P(Catch|Cavity)P(Cavity) = P(Toothache|Cavity)P(Catch|Cavity)P(Cavity) I.e., 2 + 2 + 1 = 5 independent numbers (first and second steps remove two). Else 8-1=7. The gain is bigger the more the combinations. Paolo Turrini Risk and Decisions Agent-based Systems Conditional independence contd. In most cases, the use of conditional independence reduces the size of the representation of the joint distribution from exponential in n to linear in n. Conditional independence is our most basic and robust form of knowledge about uncertain environments. Paolo Turrini Risk and Decisions Agent-based Systems The Wumps World Paolo Turrini Risk and Decisions Agent-based Systems The Wumpus World Paolo Turrini Risk and Decisions Agent-based Systems The Wumpus World Paolo Turrini Risk and Decisions Agent-based Systems The Wumpus World Paolo Turrini Risk and Decisions Agent-based Systems The Wumpus World Paolo Turrini Risk and Decisions Agent-based Systems The Wumpus World Paolo Turrini Risk and Decisions Agent-based Systems The Wumpus World Paolo Turrini Risk and Decisions Agent-based Systems The Wumpus World Paolo Turrini Risk and Decisions Agent-based Systems The Wumpus World Paolo Turrini Risk and Decisions Agent-based Systems The Wumpus World Paolo Turrini Risk and Decisions Agent-based Systems Wumpus World Paolo Turrini Risk and Decisions Agent-based Systems Wumpus World Assuming that pits can be in a square with a probability of 0.2... Paolo Turrini Risk and Decisions Agent-based Systems Wumpus World Assuming that pits can be in a square with a probability of 0.2... Pij = true iff [i, j] contains a pit Bij = true iff [i, j] is breezy Paolo Turrini Risk and Decisions Agent-based Systems Specifying the probability model Paolo Turrini Risk and Decisions Agent-based Systems Specifying the probability model Include only B1,1, B1,2, B2,1 in the probability model! Paolo Turrini Risk and Decisions Agent-based Systems Specifying the probability model Include only B1,1, B1,2, B2,1 in the probability model! The full joint distribution is P(P1,1, . . . , P4,4, B1,1, B1,2, B2,1) Paolo Turrini Risk and Decisions Agent-based Systems Specifying the probability model Include only B1,1, B1,2, B2,1 in the probability model! The full joint distribution is P(P1,1, . . . , P4,4, B1,1, B1,2, B2,1) Apply product rule: P(B1,1, B1,2, B2,1 | P1,1, . . . , P4,4)P(P1,1, . . . , P4,4) (Do it this way to get P(Effect|Cause).) Paolo Turrini Risk and Decisions Agent-based Systems Specifying the probability model Include only B1,1, B1,2, B2,1 in the probability model! The full joint distribution is P(P1,1, . . . , P4,4, B1,1, B1,2, B2,1) Apply product rule: P(B1,1, B1,2, B2,1 | P1,1, . . . , P4,4)P(P1,1, . . . , P4,4) (Do it this way to get P(Effect|Cause).) First term: 1 if pits are adjacent to breezes, 0 otherwise Paolo Turrini Risk and Decisions Agent-based Systems Specifying the probability model Include only B1,1, B1,2, B2,1 in the probability model! The full joint distribution is P(P1,1, . . . , P4,4, B1,1, B1,2, B2,1) Apply product rule: P(B1,1, B1,2, B2,1 | P1,1, . . . , P4,4)P(P1,1, . . . , P4,4) (Do it this way to get P(Effect|Cause).) First term: 1 if pits are adjacent to breezes, 0 otherwise Second term: pits are placed randomly, probability 0.2 per square: Paolo Turrini Risk and Decisions Agent-based Systems Specifying the probability model Include only B1,1, B1,2, B2,1 in the probability model! The full joint distribution is P(P1,1, . . . , P4,4, B1,1, B1,2, B2,1) Apply product rule: P(B1,1, B1,2, B2,1 | P1,1, . . . , P4,4)P(P1,1, . . . , P4,4) (Do it this way to get P(Effect|Cause).) First term: 1 if pits are adjacent to breezes, 0 otherwise Second term: pits are placed randomly, probability 0.2 per square: P(P for n pits. ,...,P ) = Π4,4 P(P ) = 0.2n ×0.816−n i ,j = 1,1 i ,j 1,1 4,4 Paolo Turrini Risk and Decisions Agent-based Systems Observations and query We know the following facts: b = ¬b1,1 ∧ b1,2 ∧ b2,1 Paolo Turrini Risk and Decisions Agent-based Systems Observations and query We know the following facts: b = ¬b1,1 ∧ b1,2 ∧ b2,1 explored = ¬p1,1 ∧ ¬p1,2 ∧ ¬p2,1 Paolo Turrini Risk and Decisions Agent-based Systems Observations and query We know the following facts: b = ¬b1,1 ∧ b1,2 ∧ b2,1 explored = ¬p1,1 ∧ ¬p1,2 ∧ ¬p2,1 Query is P(P1,3|explored, b) Paolo Turrini Risk and Decisions Agent-based Systems Observations and query We know the following facts: b = ¬b1,1 ∧ b1,2 ∧ b2,1 explored = ¬p1,1 ∧ ¬p1,2 ∧ ¬p2,1 Query is P(P1,3|explored, b) Define Unexplored = Pij s other than P1,3 and Explored Paolo Turrini Risk and Decisions Agent-based Systems Complexity For inference by enumeration, we have P(P1,3|explored, b) = αΣunexplored P(P1,3, unexplored, explored, b) Paolo Turrini Risk and Decisions Agent-based Systems Complexity For inference by enumeration, we have P(P1,3|explored, b) = αΣunexplored P(P1,3, unexplored, explored, b) There are 12 unknown squares Paolo Turrini Risk and Decisions Agent-based Systems Complexity For inference by enumeration, we have P(P1,3|explored, b) = αΣunexplored P(P1,3, unexplored, explored, b) There are 12 unknown squares The summation contains 212 = 4096 terms Paolo Turrini Risk and Decisions Agent-based Systems Complexity For inference by enumeration, we have P(P1,3|explored, b) = αΣunexplored P(P1,3, unexplored, explored, b) There are 12 unknown squares The summation contains 212 = 4096 terms In general the summation grows exponentiatlly with the number of squares! Paolo Turrini Risk and Decisions Agent-based Systems Complexity For inference by enumeration, we have P(P1,3|explored, b) = αΣunexplored P(P1,3, unexplored, explored, b) There are 12 unknown squares The summation contains 212 = 4096 terms In general the summation grows exponentiatlly with the number of squares! And now? Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence Basic insight: observations are conditionally independent of other hidden squares given neighbouring hidden squares Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence Basic insight: observations are conditionally independent of other hidden squares given neighbouring hidden squares Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence Basic insight: observations are conditionally independent of other hidden squares given neighbouring hidden squares Define Unexplored = Fringe ∪ Other Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence Basic insight: observations are conditionally independent of other hidden squares given neighbouring hidden squares Define Unexplored = Fringe ∪ Other P(b|P1,3, Explored, Unexplored) = P(b|P1,3, Explored, Fringe) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence Basic insight: observations are conditionally independent of other hidden squares given neighbouring hidden squares Define Unexplored = Fringe ∪ Other P(b|P1,3, Explored, Unexplored) = P(b|P1,3, Explored, Fringe) Manipulate query into a form where we can use this! Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence P(P1,3|explored, b) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence P(P1,3|explored, b) = α 􏰝unexplored P(P1,3, unexplored, explored, b) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence P(P1,3|explored, b) = α 􏰝unexplored P(P1,3, unexplored, explored, b) Inference by enumeration Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝unexplored P(P1,3,unexplored,explored,b) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝unexplored P(P1,3,unexplored,explored,b) = α 􏰝unexplored P(b|explored, P1,3, unexplored)× ×P(P1,3 , explored , unexplored ) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝unexplored P(P1,3,unexplored,explored,b) = α 􏰝unexplored P(b|explored, P1,3, unexplored)× ×P(P1,3 , explored , unexplored ) Product rule Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝unexplored P(b|P1,3,unexplored,explored)P(P1,3,unexplored,explored) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝unexplored P(b|P1,3,unexplored,explored)P(P1,3,unexplored,explored) = α 􏰝fringe 􏰝other P(b|explored, P1,3, fringe, other)× ×P(P1,3 , explored , fringe , other ) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝unexplored P(b|P1,3,unexplored,explored)P(P1,3,unexplored,explored) = α 􏰝fringe 􏰝other P(b|explored, P1,3, fringe, other)× ×P(P1,3 , explored , fringe , other ) Distinguishing the unknown Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringe 􏰝other P(b|explored,P1,3,fringe,other)× ×P(P1,3 , explored , fringe , other ) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringe 􏰝other P(b|explored,P1,3,fringe,other)× ×P(P1,3 , explored , fringe , other ) = α 􏰝fringe 􏰝other P(b|explored, P1,3, fringe)× ×P(P1,3 , explored , fringe , other ) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringe 􏰝other P(b|explored,P1,3,fringe,other)× ×P(P1,3 , explored , fringe , other ) = α 􏰝fringe 􏰝other P(b|explored, P1,3, fringe)× ×P(P1,3 , explored , fringe , other ) Conditional Independence Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringe 􏰝other P(b|explored,P1,3,fringe)× ×P(P1,3 , explored , fringe , other ) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringe 􏰝other P(b|explored,P1,3,fringe)× ×P(P1,3 , explored , fringe , other ) = α􏰝fringeP(b|explored,P1,3,fringe)× × 􏰝other P(P1,3 , explored , fringe , other ) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringe 􏰝other P(b|explored,P1,3,fringe)× ×P(P1,3 , explored , fringe , other ) = α􏰝fringeP(b|explored,P1,3,fringe)× × 􏰝other P(P1,3 , explored , fringe , other ) Pushing the sums inwards Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringeP(b|explored,P1,3,fringe)× 􏰝other P(P1,3 , explored , fringe , other ) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringeP(b|explored,P1,3,fringe)× 􏰝other P(P1,3 , explored , fringe , other ) = α 􏰝fringe P(b|explored, P1,3, fringe)× 􏰝other P(P1,3)P(explored)P(fringe)P(other) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringeP(b|explored,P1,3,fringe)× 􏰝other P(P1,3 , explored , fringe , other ) = α 􏰝fringe P(b|explored, P1,3, fringe)× 􏰝other P(P1,3)P(explored)P(fringe)P(other) Independence Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringe P(b|explored,P1,3,fringe)× ×􏰝other P(P1,3)P(explored)P(fringe)P(other) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringe P(b|explored,P1,3,fringe)× ×􏰝other P(P1,3)P(explored)P(fringe)P(other) = αP(explored)P(P1,3)× ×􏰝fringe P(b|explored,P1,3,fringe)P(fringe)􏰝other P(other) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α􏰝fringe P(b|explored,P1,3,fringe)× ×􏰝other P(P1,3)P(explored)P(fringe)P(other) = αP(explored)P(P1,3)× ×􏰝fringe P(b|explored,P1,3,fringe)P(fringe)􏰝other P(other) Reordering and pushing sums inwards Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α P (explored )P(P1,3 )× ×􏰝fringe P(b|explored,P1,3,fringe)P(fringe)􏰝other P(other) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α P (explored )P(P1,3 )× ×􏰝fringe P(b|explored,P1,3,fringe)P(fringe)􏰝other P(other) = α′ P(P1,3) 􏰝fringe P(b|explored, P1,3, fringe)P(fringe) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence α P (explored )P(P1,3 )× ×􏰝fringe P(b|explored,P1,3,fringe)P(fringe)􏰝other P(other) = α′ P(P1,3) 􏰝fringe P(b|explored, P1,3, fringe)P(fringe) Simplifying Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence contd. P(b|explored, P1,3, fringe) Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence contd. P(b|explored, P1,3, fringe) = 1 when the frontier is consistent with the observations = 0 otherwise Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence contd. P(b|explored, P1,3, fringe) = 1 when the frontier is consistent with the observations = 0 otherwise We can sum over the possible configurations for the frontier variables that are consistent with the known facts. Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence contd. P(P1,3|explored, b)= Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence contd. P(P1,3|explored, b)= α′ ⟨0.2(0.04 + 0.16 + 0.16), 0.8(0.04 + 0.16)⟩ Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence contd. P(P1,3|explored, b)= α′ ⟨0.2(0.04 + 0.16 + 0.16), 0.8(0.04 + 0.16)⟩ ≈ ⟨0.31, 0.69⟩ Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence contd. P(P1,3|explored, b)= α′ ⟨0.2(0.04 + 0.16 + 0.16), 0.8(0.04 + 0.16)⟩ ≈ ⟨0.31, 0.69⟩ P(P2,2|explored, b) ≈ ⟨0.86, 0.14⟩ Paolo Turrini Risk and Decisions Agent-based Systems Using conditional independence contd. P(P1,3|explored, b)= α′ ⟨0.2(0.04 + 0.16 + 0.16), 0.8(0.04 + 0.16)⟩ ≈ ⟨0.31, 0.69⟩ P(P2,2|explored, b) ≈ ⟨0.86, 0.14⟩ Paolo Turrini Risk and Decisions Agent-based Systems What we have seen Probabilities and conditional probabilities Independence and conditional independence Estimating chances of possible outcomes Paolo Turrini Risk and Decisions Agent-based Systems Coming next Combining chances and rewards Maximising the expected reward Paolo Turrini Risk and Decisions Agent-based Systems