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
αunexplored P(P1,3,unexplored,explored,b)
Paolo Turrini Risk and Decisions Agent-based Systems
Using conditional independence
αunexplored 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
αunexplored 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
αunexplored P(b|P1,3,unexplored,explored)P(P1,3,unexplored,explored)
Paolo Turrini Risk and Decisions Agent-based Systems
Using conditional independence
αunexplored 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
αunexplored 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
αfringe 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
αfringe 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
αfringe 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
αfringe other P(b|explored,P1,3,fringe)× ×P(P1,3 , explored , fringe , other )
Paolo Turrini Risk and Decisions Agent-based Systems
Using conditional independence
αfringe other P(b|explored,P1,3,fringe)× ×P(P1,3 , explored , fringe , other )
= αfringeP(b|explored,P1,3,fringe)× × other P(P1,3 , explored , fringe , other )
Paolo Turrini Risk and Decisions Agent-based Systems
Using conditional independence
αfringe other P(b|explored,P1,3,fringe)× ×P(P1,3 , explored , fringe , other )
= αfringeP(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
αfringeP(b|explored,P1,3,fringe)× other P(P1,3 , explored , fringe , other )
Paolo Turrini Risk and Decisions Agent-based Systems
Using conditional independence
αfringeP(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
αfringeP(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
αfringe 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
αfringe 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
αfringe 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