1. Uncertainty
Chapter 13
Copyright, 1996 © Dale Carnegie & Associates, Inc.

2. Uncertainty
Evolution of an intelligent agent: problem solving, planning, uncertainty
Dealing with uncertainty is an unavoidable problem in reality.
An agent must act under uncertainty.
To make decision with uncertainty, we need
Probability theory
Utility theory
Decision theory
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3. Sources of uncertainty
No access to the whole truth
No categorical answer
Incompleteness
The qualification problem - impossible to explicitly enumerate all conditions
Incorrectness of information about conditions
The rational decision depends on both the relative importance of various goals and the likelihood of its being achieved.
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4. Handling uncertain knowledge
Difficulties in using FOL to cope with UK
A dental diagnosis system using FOL
Symptom (p, Toothache) =>Disease (p, Cavity)
Disease (p, Cavity) => Symptom (p, Toothache)
Are they correct?
Reasons for handling uncertain knowledge
Laziness - too much work! How to avoid it?
Theoretical ignorance - we don’t know everything
Practical ignorance - we don’t want to include all
Represent UK with a degree of belief
The tool for handling UK is probability theory
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5. Probability, belief of the truth of a sentence
Probability provides a way of summarizing the uncertainty that comes from our laziness and ignorance - how wonderful it is!
Probability, belief of the truth of a sentence
1 - true, 0 - false, 0<P<1 - intermediate degrees of belief in the truth of the sentence
Degree of truth (fuzzy logic) vs. degree of belief
Alternatives to probability theory?
Yes, to be discussed in later chapters.
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6. All probability statements must indicate the evidence with respect to which the probability is being assessed.
Prior or unconditional probability before evidence is obtained
Posterior or conditional probability after new evidence is obtained
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7. Uncertainty & rational decisions
Without uncertainty, decision making is simple - achieving the goal or not
With uncertainty, it becomes uncertain - three plans A90, A120 and A1440
We first need to have preferences between the different possible outcomes of the plans
Utility theory is used to represent and reason with preferences.
A - Airport
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8. Rationality Decision Theory = Probability T + Utility T
Maximum Expected Utility Principle defines rationality
An agent is rational if and only if it chooses the action that yields the highest utility, averaged over all possible outcomes of the action
A decision-theoretic agent (pseudocode, Fig 13.1, p 466)
Is it any different from other agents we learned?
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9. Basic probability notation
Prior probability
Proposition - P(Sunny)
Random variable - P(Weather=Sunny)
Boolean, discrete, and continuous random variables
Each RV has a domain (sunny,rain,cloudy,snow)
Probability distribution P(weather) = <.7,.2,.08,.02>
Joint probability P(A^B)
probabilities of all combinations of the values of a set of Random Variables
more later
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10. Conditional probability
P(A|B) = P(A^B)/P(B)
Product rule - P(A^B) = P(A|B)P(B)
Probabilistic inference does not work like logical inference
“P(A|B)=0.6” != “when B is true, P(A) is 0.6”
P(A) is always a prior.
For P(A|B), B is the only available evidence.
When C is available, P(A|B,C) may have little relation to P(A|B).
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11. The axioms of probability
All probabilities are between 0 and 1
Necessarily true (valid) propositions have probability 1, false (unsatisfiable) 0
The probability of a disjunction
P(AvB)=P(A)+P(B)-P(A^B)
A Venn diagram illustration
P(a v !a) is 1
Ex: Deriving the rule of
Negation from P(a v !a)
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12. The joint probability distribution
Joint completely specifies an agent’s probability assignments to all propositions in the domain
A probabilistic model consists of a set of random variables (X1, …,Xn).
An atomic event is an assignment of particular values to all the variables
Given Boolean random variables A and B, what are atomic events?
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13. Joint probabilities An example of two Boolean variables
Observations: mutually exclusive and
collectively exhaustive
What are P(Cavity), P(Cavity v Toothache),
P(Cavity|Toothache)?
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14. Joint (2)
If there is a Joint distribution, we can read off any probability we need. Is it true? How?
Discuss next
Impractical to specify all the entries for the Joint over n Boolean variables.
Sidestep the Joint and work directly with conditional probability
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15. Inference using full joint distributions
Marginal probability (Joint in Fig 13.3)
P(cavity) =
Maginalization – summing out all the variables other than cavity
P(Y) = z P(Y,z)
Conditioning – a variant of maginalization using the product rule
P(Y) = z P(Y|z)P(z)
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16. Normalization Method 1 using P(t) to nomalize
P(cavity|toothache) = P(c^t)/P(t)
P(!cavity|toothache) = P(!c^t)/P(t)
Method 2 using α and leaving P(t) out
P(cavity|toothache) = αP(cavity,toothache)
= α[P(cavity, T, catch) + P(cavity, T, !catch)]
= α[<0.108,0.016> + <0.012,0.064>] = α<0.12,0.08>
What is α?
P(t) is the same
Joint:
Cav vs.
!Cav
Fig 13.3
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17. Independence P(toothache, catch, cavity, weather)
A total of 32 entries, given W has 4 values
How is one’s tooth problem related to weather?
P(T,Ch,Cy,W=cloudy) = P(W=Clo|T...)P(T…)?
Whose tooth problem can influence our weather?
P(W=Clo|T…) = P(W=Clo)
Hence, P(T,Ch,Cy,W=clo) = P(W=Clo)P(T…)
How many joint distribution tables? Two - (4, 8)
Independence between X and Y means
P(X|Y) = P(X) or P(Y|X) = P(Y) or P(XY) = P(X)P(Y)
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18. Bayes’ rule Deriving the rule via the product rule
P(B|A) = P(A|B)P(B)/P(A)
A more general case is P(X|Y) = P(Y|X)P(X)/P(Y)
Bayes’ rule conditionalized on evidence E
P(X|Y,E) = P(Y|X,E)P(X|E)/P(Y|E)
Applying the rule to medical diagnosis
meningitis (P(M)=1/50,000)), stiff neck (P(S)=1/20), P(S|M)=0.5, what is P(M|S)?
Why is this kind of inference useful?
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19. Applying Bayes’ rule Relative likelihood
Comparing the relative likelihood of meningitis and whiplash, given a stiff neck, which is more likely?
P(M|S)/P(W|S) = P(S|M)P(M)/P(S|W)P(W)
Avoiding direct assessment of the prior (P(S))
P(M|S) =? P(!M|S) =? And P(M|S) + P(!M|S) = 1,
P(S) = ? P(S|!M) = ?
P (S) = m P(Sm) = m P(S|m) P(m) = P(S|M)P(M) + P(S|!M)P(!M)
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20. Using Bayes’ rule Combining evidence Bayesian updating
from P(Cavity|Toothache) and P(Cavity|Catch) to P(Cavity|Toothache,Catch)
Bayesian updating
from P(Cavity|T)=P(Cavity)P(T|Cavity)/P(T)
P(A|B) = P(B|A)P(A)/P(B)
to P(Cavity|T,Catch)=P(Catch|T,Cavity)/P(Catch|T)
P(A|B,C) = P(B|A,C)P(A|C)/P(B|C)
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21. Recall that independent events A, B
P(B|A)=P(B), P(A|B)=P(A), P(A,B)=P(A)P(B)
Conditional independence (X and Y are ind given Z)
P(X|Y,Z)=P(X|Z) and P(Y|X,Z)=P(Y|Z)
P(XY|Z)=P(X|Z)P(Y|Z) derived from absolute indepence
Given Cavity, Toothache and Catch are indpt
P(T,Ch,Cy) = P(T,Ch|Cy)P(Cy) = P(T|Cy)P(Ch|Cy)P(Cy)
One large table is decomposed into 3 smaller tables: 23-1 vs. 5 (= 2*(21-1)+2*(21-1)+(21-1))
T|Cy
T|!Cy Cy
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22. Independence, decomposition, Naïve Bayes
If all n symptoms are conditionally indpt given Cavity, the size of the representation grows as O(n) instead of O(2n)
The decomposition of large probabilistic domains into weakly connected subsets via conditional independence is one important development in modern AI
Naïve Bayes model (Cause and Effects)
P(Cause,E1,…,En) = P(Cause)  P(Ei|Cause)
An amazingly successful classifier can be built
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23. Where do probabilities come from?
There are three positions:
The frequentist - numbers can come only from experiments
The objectivist - probabilities are real aspects of the universe
The subjectivist - characterizing an agent’s belief
The reference class problem – what constitutes the reference class - intrusion of subjectivity
A frequentist doctor wants to consider similar patients
How similar two patients are?
Laplace’s principle of indifference
Propositions that are syntactically “symmetric” w.r.t. the evidence should be accorded equal probability
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24. Summary Uncertainty exists in the real world.
It is good (as it allows for laziness) and bad (since we need new tools)
Priors, posteriors, and joint
Bayes’ rule - the base of Bayesian Inference
Conditional independence allows Bayesian updating to work effectively with many pieces of evidence.
But ...
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