1. Uncertainty in Environments
Comp3710 Artificial Intelligence
Computing Science
Thompson Rivers University

2. Course Outline Part I – Introduction to Artificial Intelligence
Part II – Classical Artificial Intelligence
Part III – Machine Learning
Introduction to Machine Learning
Neural Networks
Probabilistic Reasoning and Bayesian Belief Networks
Artificial Life: Learning through Emergent Behavior
Part IV – Advanced Topics
TRU-COMP3710
Uncertainty

3. Learning Outcomes
Relate a proposition to probability, i.e., random variables.
Give the sample space for a set of random variables.
Use one of Komogorov’s 3 axioms
P(A  B) = P(A) + P(B) – P(A  B)
Give the joint probability distribution for a set of random variables
Compute probabilities given a joint probability distribution
Give the joint probability distribution for a set of random variables that have conditional independence relations.
Compute the diagnostic probability from causal probabilities.
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4. Outline How to deal with uncertainty in environments Probability
Syntax and Semantics
Inference
Independence and Bayes' Rule
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5. 1. Uncertainty in environments
An example of decision making problem:
Let action At = leave for airport t minutes before flight.
Will At get me there on time?
[Q] How to decide?
Problems: Application environments are not deterministic and unpredictable. Many hidden factors.
partial observability (road state, other drivers' plans, etc.)
noisy sensors (traffic reports)
uncertainty in action outcomes (flat tire, etc.)
immense complexity of modeling and predicting traffic
Hence a purely logical approach either
risks falsehood: “A25 will get me there on time”, or
leads to conclusions that are too weak for decision making:
“A25 will get me there on time if there's no accident on the bridge and it doesn't rain and my tires remain intact etc etc.”
(A1440 might reasonably be said to get me there on time but I'd have to stay overnight in the airport …)
Uncertainty

6. Methods for handling uncertainty
Topics
Methods for handling uncertainty
[Q] How to handle uncertainty?
[Q] Default logic?
Default assumptions: My car does not have a flat tire. ...
A25 works unless contradicted by evidence.
But issues: What assumptions are reasonable?
[Q] Can we use fuzzy logic?
Probability – uncertainty/certainty
Model of degree of belief
Given the available evidence or knowledge – environments,
A25 will get me there on time with probability 0.04.
Fuzzy logic – ambiguity/clearness
Degree of truth – value (observed data/facts)
E.g., my age
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7. 2. Probability
Probability is a way of expressing knowledge or belief that an event will occur or has occurred. E.g., …
Probabilistic assertions summarize effects of hidden factors, i.e.,
laziness: failure to enumerate exceptions, qualifications, etc.
ignorance: lack of relevant facts, initial conditions, etc.
Subjective or Bayesian probability:
Probabilities can relate propositions to agent's own state of knowledge (or conditions).
e.g., P(A25 | no reported accidents) = 0.06
Probabilities of propositions change with new evidence:
e.g., P(A25 | no reported accidents, 5 a.m.) = 0.15
Hence probability theory can be a good tool.
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8. Making decisions under uncertainty
Topics
Making decisions under uncertainty
Suppose I believe the following:
P(A25 gets me there on time | …) = 0.04
P(A90 gets me there on time | …) = 0.70
P(A120 gets me there on time | …) = 0.95
P(A1440 gets me there on time | …) =
[Q] Which action to choose?
Depends on my preferences for missing flight vs. time spent waiting, etc.
Utility (the quality of being useful) theory is used to represent and infer preferences.
Decision theory = probability theory + utility theory
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9. 3. Syntax and Semantics
Sample space, event, probability, probability distribution
Random variable and proposition
Prior probability (aka unconditional probability)
Posterior probability (aka conditional probability)
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10. Sample Space and Probability
Definition. Begin with a set Ω – the sample space of the same type atomic events
E.g., 6 possible rolls of a die -> Ω = {???}
ω in Ω is a sample point / possible world / atomic event
Definition. A probability space or probability distribution or probability model is a sample space with an assignment P(ω) for every ω in Ω s.t.
0 ≤ P(ω) ≤ 1
Σω P(ω) = 1
E.g., P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6. [Q] Sample space? [Q] Probability distribution?
Definition. An event A can be considered as a subset of Ω.
P(A) = Σω in A P(ω)
E.g., P(die roll < 4) = P(1) + P(2) + P(3) = 1/6 + 1/6 + 1/6 = 1/2
Surface 1 2 3 4 5 6
Probability
1/6
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11. 3.
Axioms of probability
Kolmogorov’s axioms: For any propositions (events) A, B
0 ≤ P(A) ≤ 1
P(true) = 1 and P(false) = 0
P(A  B) = P(A) + P(B) – P(A  B)
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12. Random Variable and Proposition
Definition. Basic element: random variable, representing a sample space
A function from sample points to some range, e.g., the reals or Booleans
E.g., Odd: N -> {true, false} Odd(1) = true
Boolean random variables
E.g., Cavity (do I have a cavity?); [Q] What is the sample space?
Discrete random variables (finite or infinite)
E.g., Weather is one of <sunny, rainy, cloudy, snow>; [Q] What is the sample space?
Continuous random variables (bounded or unbounded)
E.g., RoomTemperature < 0.22; [Q] What is the sample space?
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13. Proposition (how to relate to probability?)3.
Proposition (how to relate to probability?)
Think of a proposition as the event where the proposition is true.
E.g., I have a toothache; The weather is sunny and there is no cavity.
[Q] How to relate to probability, i.e., random variables?
Similar to propositional logic: possible worlds defined by assignment of values to random variables.
A sample point is a value of a set of random variables.
A sample space is the Cartesian product of the ranges of random variables.
[Q] What is the sample space for Cavity and Weather?
Elementary proposition constructed by assignment of a value to a random variable:
E.g., Weather = sunny; Cavity = false (can be abbreviated as cavity)
Complex propositions formed from elementary propositions and standard logical connectives
E.g., Weather = sunny  Cavity = false
Proposition = conjunction/disjunction of atomic events
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14. Prior and Posterior Probability
Prior (aka unconditional) probability
Posterior (aka conditional) probability
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15. Prior probability Prior or unconditional probabilities of propositions
E.g., P(Cavity = true) = 0.1 and P(Weather = sunny) = 0.72 correspond to belief prior to arrival of any (new) evidence.
Probability distribution gives values for all possible assignments:
P(Weather) = <0.72, 0.1, 0.08, 0.1> (normalized, i.e., sums to 1)
Joint probability distribution for a set of random variables gives the probability of every atomic event on those random variables.
P(Weather, Cavity) = a 4 × 2 matrix of values:
Weather = sunny rainy cloudy snow
Cavity = true
[Q] What is the sum of all the values above?
Every question about a domain can be answered with the joint probability distribution. [Q] Any problem? Size of the table.
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16. P(Weather, Cavity) = a 4 × 2 matrix of values:
Weather = sunny rainy cloudy snow
Cavity = true
Cavity = false
[Q] What is the probability of cloudy?
[Q] What is the probability of snow and no-cavity?
[Q] What is the probability of sunny or cavity?
P(W=sunny)  P(C=true)
= P(W=sunny) + P(C=true) ??? ???
= ( ) + ( ) – ???
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17. Prior probability – continuous variables
In general, express probability distribution as parameterized function, called a probability density function (pdf), of value
E.g., P(X=x) = U[18,26](x) = 0.125
P is a density; integrates to 1
0.125 18 dx 26
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18. Example of Normal distribution, or also called Gaussian distribution
Probabilities
[This image is
from Wikipedia.org]
Events
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19. Interesting facts: 3. P(A) = 1 – P(~A)
P(A) = P(A  B) + P(A  ~B) for any B
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20. Conditional probability
Posterior or conditional probabilities
i.e., given that toothache is all I know, (it is not like if toothache,) 60% chance of cavity.
Another example, P(toothache | cavity) = 0.8
New evidence/observation/fact may be irrelevant, allowing simplification, e.g., when Cavity and Weather are irrelevant,
P(cavity | toothache, sunny) = P(cavity | toothache) = 0.6
This kind of inference, sanctioned by domain knowledge, is crucial.
Very interesting examples: [Q] The stiff neck is one symptom of meningitis; the probability of having meningitis with stiff neck?
[Q] A car won’t start.
What is the probability of having a bad starter?
What is the probability of having a bad battery?
Uncertainty

21. Definition of conditional probability:
P(a | b) = P(a  b) / P(b) if P(b) > 0
Product rule gives an alternative formulation:
P(a  b) = P(a) P(b | a) = P(b) P(a | b)
Chain rule is derived by successive application of product rule:
P(a  b  c) = ???
= P(a  b) P(c | a  b)
= P(a) P(b | a) P(c | a  b)
P(X1,…, Xn) = P(X1,..., Xn-1) P(Xn | X1,..., Xn-1)
= P(X1,..., Xn-2) P(Xn-1 | X1,..., Xn-2) P(Xn | X1,..., Xn-1)
= …
= P(X1) P(X2 | X1) P(X3 | X1, X2) P(X4 | X1, X2, X3) …
= i=1 P(Xi | X1,…, Xi-1) will be used in Inference later! a n
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22. 3.
Topics
Interesting facts:
P(A|B) = 1 – P(~A|B)
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23. 4. Inference by Enumeration
Start with the joint probability distribution: Toothache (I have a toothache), Cavity (meaning?), Catch (catch probe?).
[Q] What are random variables here?
[Q] What is the sum of probabilities of all the atomic events?
[Q] How can you obtain the above probability distribution?
For any proposition φ, the probability is the sum of the atomic events where it is true:
P(φ) = Σω:ω╞φ P(ω)
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24. For any proposition φ, sum the atomic events where it is true:
Start with the joint probability distribution: Toothache, Cavity, Catch
For any proposition φ, sum the atomic events where it is true:
P(φ) = Σω:ω╞φ P(ω)
[Q] P(toothache) = = 0.2
[Q] P(cavity  toothache) = ???
[Q] P(cavity  toothache) = ???
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25. Can also compute conditional probabilities: Very important!
Start with the joint probability distribution: Toothache, Cavity, Catch
Can also compute conditional probabilities: Very important!
P(cavity | toothache) = P(cavity  toothache)
P(toothache) =
= 0.4
[Q] P(cavity | toothache) = ??? P(cavity | toothache) = ???
[Q] P(cavity | catch) = ???
[Q] P(cavity | toothache) ???
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26. Denominator can be viewed as a normalization constant α
P(Cavity | toothache) = P(Cavity, toothache) / P(toothache)
= α, P(Cavity, toothache)
= α, [P(Cavity, toothache, catch) + P(Cavity, toothache, catch)]
= α, [<0.108, 0.016> + <0.012, 0.064>]
= α, <0.12, 0.08> = <0.6, 0.4>
[Q] What is α?
General idea: compute distribution on query variable by fixing evidence variables and summing over hidden variables
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27. Obvious problems: In other words, too much. [Q] How to reduce? Topics
For n random variables,
Worst-case time complexity O(dn) where d is the largest arity
Space complexity O(dn) to store the joint distribution
How to find the numbers for O(dn) entries?
In other words, too much.
[Q] How to reduce?
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28. 5. Independence & Bayes’ Rule
Conditional independence
Bayes’ rule
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29. 5.
Independence
A and B are independent iff P(A|B) = P(A) or P(B|A) = P(B)
or P(A,B) = P(A) P(B)
P(Toothache, Catch, Cavity, Weather) = P(Toothache, Catch, Cavity) × P(Weather)
In the above example, 32 (2*2*2*4) entries reduced to 12 (2*2*2 + 4).
Absolute independence is powerful but rare.
Dentistry is a large field with hundreds of variables, none of which are independent. [Q] What do we have to do?
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30. Conditional Independence
If I have a cavity, the probability that the probe catches in it doesn't depend on whether I have a toothache:
(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:
P(Catch | Toothache, Cavity) = P(Catch | Cavity)
When Catch is conditionally independent of Toothache given Cavity, equivalent statements:
P(Toothache | Catch, Cavity) = P(Toothache | Cavity)
P(Toothache, Catch | Cavity) = P(Toothache | Cavity) P(Catch | Cavity)
(similarly P(A, B) = P(A) P(B) when A and B are independent)
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31. Write out full joint distribution using chain rule:
P(Toothache, Catch, Cavity)
= P(Toothache | Catch, Cavity) P(Catch, Cavity) by the product rule
= P(Toothache | Catch, Cavity) P(Catch | Cavity) P(Cavity) by ???
= P(Toothache | Cavity) P(Catch | Cavity) P(Cavity) by ???
2 * 2 * 2 = 8 -> How ???
Cavity
Toothache
Catch
[Q] P(toothache) from the tables below???
= P(tcavity) + P(tcavity)
= P(t|c) P(c) + P(t|c) P(c) = ???
[Q] P(t|cavity) = ???
= 1 – P(t|c) = ??? ?
toothache | cavity
toothache | cavity
0.6
0.1
catch
| cavity
| cavity
0.9
0.2
cavity
0.2
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32. 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)
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.
Cavity
Toothache
Catch
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33. It is how to show the conditional independency.5.
Interesting fact:
P(A,B|C) = P(A|C) P(B|C) when A and B are conditionally independent of each other given C.
Actually it is a definition.
P(toothache  catch | cavity) = .108 / ( ) = .054
P(toothache | cavity) = ( ) / .2 = .12 / .2 = .06
P(catch | cavity) = ( ) / .2 = .09
P(toothache | cavity) P(catch | cavity) = .06 * .09 = .054
 P(toothache  catch | cavity) = P(toothache | cavity) P(catch | cavity) …
Therefore Toothache and Catch are conditionally independent given Cavity.
It is how to show the conditional independency.
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34. Bayes' Rule Product rule P(ab) = P(a | b) P(b) = P(b | a) P(a)
 Bayes' rule, or also called Bayes’ Theorem:
P(a | b) = P(b | a) P(a) / P(b)
or in distribution form
P(Y|X) = P(X|Y) P(Y) / P(X) = α P(X|Y) P(Y)
Useful for assessing diagnostic probability from causal probability:
P(Cause|Effect) = P(Effect|Cause) P(Cause) / P(Effect)
[Q] The stiff neck is one symptom of meningitis; the probability of having meningitis with stiff neck?
[Q] A car won’t start. What is the probability of having a bad starter? What is the probability of having a bad battery?
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35. Useful for assessing diagnostic probability from causal probability:
P(Cause|Effect) = P(Effect|Cause) P(Cause) / P(Effect)
[Q] When cavity, usually toothache
P(cavity | toothache) ?
[Q] E.g., let M be meningitis, S be stiff neck; the stiff neck is one symptom of meningitis; the probability of having meningitis with stiff neck?
P(s) = 0.1; P(m) = ; P(s|m) = 0.8
P(m|s) = ???
[Q] How to find P(s|m), P(m) and P(s) ???
Note: posterior probability of meningitis still very small!
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36. Useful for assessing diagnostic probability from causal probability:
P(Cause|Effect) = P(Effect|Cause) P(Cause) / P(Effect)
[Q] E.g., let B be ‘bad battery’, N be ‘no starting in cold morning’; N is one symptom of B; the probability of having bad battery when a car won’t start in cold morning?
P(n) = 0.01; P(b) = 0.05; P(n|b) = 0.8
P(b|n) = ???
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37. Bayes' rule and conditional independence5.
Bayes' rule and conditional independence
Topics
P(Cavity | toothache  catch)
= P(toothache  catch | Cavity) P(Cavity)
= P(toothache | Cavity) P(catch | Cavity) P(Cavity)
This is an example of a naïve Bayes model:
P(Cause, Effect1, … , Effectn) = P(Cause) Πi P(Effecti|Cause), where
Effects are conditionally independent of each other, given Cause.
We will use this later again!
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38. Topics
6. Summary
Probability is a rigorous formalism for uncertain knowledge.
Joint probability distribution for discrete random variables specifies probability of every atomic event.
Queries can be answered by summing over atomic events.
For nontrivial domains, we must find a way to reduce the joint size.
Independence and conditional independence provide the tools.
Uncertainty