2101INT – Principles of Intelligent Systems Lecture 9
Jan 17, 2016
2101INT – Principles of Intelligent Systems
Lecture 9
Outline – Part 1
Uncertainty Probability Syntax and Semantics Inference Independence and Bayes' Rule
Part 1 slides (2-21) are taken from those available from the AIMA website at http://aima.cs.berkeley.edu
Author recorded as Min-Yen Kan, NUS
Ch13
Uncertainty
Let action At = leave for airport t minutes before flightWill At get me there on time?
Problems:1. partial observability (road state, other drivers' plans, etc.)2. noisy sensors (traffic reports)3. uncertainty in action outcomes (flat tire, etc.)4. immense complexity of modeling and predicting traffic
Hence a purely logical approach either1. risks falsehood: “A25 will get me there on time”, or2. 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 …)
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Methods for handling uncertainty
Default or nonmonotonic logic:– Assume my car does not have a flat tire– Assume A25 works unless contradicted by evidence
Issues: What assumptions are reasonable? How to handle contradiction?
Rules with fudge factors:– A25 |→0.3 get there on time– Sprinkler |→ 0.99 WetGrass– WetGrass |→ 0.7 Rain
Issues: Problems with combination, e.g., Sprinkler causes Rain??
Probability– Model agent's degree of belief– Given the available evidence,– A25 will get me there on time with probability 0.3–
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Probability
Probabilistic assertions summarize effects of– laziness: failure to enumerate exceptions, qualifications, etc.– ignorance: lack of relevant facts, initial conditions, etc.
Subjective probability: Probabilities relate propositions to agent's own state of knowledge
e.g., P(A25 | no reported accidents) = 0.06
These are not assertions about the world
Probabilities of propositions change with new evidence:e.g., P(A25 | no reported accidents, 5 a.m.) = 0.15
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p463
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 | …) = 0.9999
Which action to choose?Depends on my preferences for missing flight vs. time spent waiting, etc.
– Utility theory is used to represent and infer preferences– Decision theory = probability theory + utility theory–
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p465
Syntax
Basic element: random variable
Similar to propositional logic: possible worlds defined by assignment of values to random variables.
Boolean random variablese.g., Cavity (do I have a cavity?)
Discrete random variablese.g., Weather is one of <sunny,rainy,cloudy,snow>
Domain values must be exhaustive and mutually exclusive
Elementary proposition constructed by assignment of a value to a
Complex propositions formed from elementary propositions and standard logical connectives e.g., Weather = sunny Cavity = false
random variable: e.g., Weather = sunny, Cavity = false (abbreviated as cavity)
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Syntax
Atomic event: A complete specification of the state of the world about which the agent is uncertainE.g., if the world consists of only two Boolean variables Cavity
and Toothache, then there are 4 distinct atomic events:Cavity = false Toothache = falseCavity = false Toothache = trueCavity = true Toothache = falseCavity = true Toothache = true
Atomic events are mutually exclusive and exhaustive
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p468
Axioms of probability
For any propositions A, B– 0 ≤ P(A) ≤ 1– P(true) = 1 and P(false) = 0– P(A B) = P(A) + P(B) - P(A B)–
p471
Prior probability
Prior or unconditional probabilities of propositionse.g., P(Cavity = true) = 0.2 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 variablesP(Weather,Cavity) = a 4 × 2 matrix of values:
Weather = sunny rainy cloudy snow Cavity = true 0.144 0.02 0.016 0.02Cavity = false 0.576 0.08 0.064 0.08
Every question about a domain can be answered by the joint distribution
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Conditional probability
Conditional or posterior probabilitiese.g., P(cavity | toothache) = 0.8i.e., given that toothache is all I know
(Notation for conditional distributions:P(Cavity | Toothache) = 2-element vector of 2-element vectors)
If we know more, e.g., cavity is also given, then we haveP(cavity | toothache,cavity) = 1
New evidence may be irrelevant, allowing simplification, e.g.,P(cavity | toothache, sunny) = P(cavity | toothache) = 0.8
This kind of inference, sanctioned by domain knowledge, is crucial
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p470
Conditional probability
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 | b) P(b) = P(b | a) P(a)
A general version holds for whole distributions, e.g.,P(Weather,Cavity) = P(Weather | Cavity) P(Cavity)
(View as a set of 4 × 2 equations, not matrix mult.)
Chain rule is derived by successive application of product rule: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) = … = πi= 1^n P(Xi | X1, … ,Xi-1)
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Inference by enumeration
Start with the joint probability distribution:
For any proposition φ, sum the atomic events where it is true: P(φ) = Σω:ω╞φ P(ω)
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Inference by enumeration
Start with the joint probability distribution:
For any proposition φ, sum the atomic events where it is true: P(φ) = Σω:ω╞φ P(ω)
P(toothache) = 0.108 + 0.012 + 0.016 + 0.064 = 0.2
Start with the joint probability distribution:
Can also compute conditional probabilities:
P(cavity | toothache) = P(cavity toothache)
P(toothache)
= 0.016+0.064
0.108 + 0.012 + 0.016 + 0.064
= 0.4
Inference by enumeration
Normalization
Denominator can be viewed as a normalization constant α
P(Cavity | 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>
General idea: compute distribution on query variable by fixing evidence variables and summing over hidden variables
Independence
A and B are independent iffP(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)
32 entries reduced to 12=(8+4); for n independent biased coins, O(2n) →O(n)
Absolute independence powerful but rare
Dentistry is a large field with hundreds of variables, none of which are independent. What to do?
p477
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)
Equivalent statements:P(Toothache | Catch, Cavity) = P(Toothache | Cavity)P(Toothache, Catch | Cavity) = P(Toothache | Cavity) P(Catch | Cavity)
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Bayes' Rule
Product rule P(ab) = P(a | b) P(b) = P(b | a) P(a) Bayes' rule: P(a | b) = P(b | a) P(a) / P(b)
or in distribution form P(Y|X) = P(X|Y) P(Y) / P(X) = αP(X|Y) P(Y)
Useful for assessing diagnostic probability from causal probability:– P(Cause|Effect) = P(Effect|Cause) P(Cause) / P(Effect)
– E.g., let M be meningitis, S be stiff neck:P(m|s) = P(s|m) P(m) / P(s) = 0.8 × 0.0001 / 0.1 = 0.0008
– Note: posterior probability of meningitis still very small!–
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Bayes' Rule and conditional independence
P(Cavity | toothache catch) = αP(toothache catch | Cavity) P(Cavity)
= αP(toothache | Cavity) P(catch | Cavity) P(Cavity)
This is an example of a naïve Bayes model:P(Cause,Effect1, … ,Effectn) = P(Cause) πiP(Effecti|Cause)
Total number of parameters is linear in n
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Summary
Probability is a rigorous formalism for uncertain knowledge
Joint probability distribution 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
Outline – Part 2
Non-monotonic logics Uncertainty and temporal knowledge
Non-monotonicity
Humans are not first-order theorem provers – everyday we are faced with making decisions based on uncertain and incomplete information
Classical logics do not handle such scenarios well Any new information, or facts, added to the knowledge
base are expected to be consistent with the existing facts
When new information can contradict previously drawn conclusions, this is said to be non-monotonicity
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The non-monotonicity problem
Humans are not first-order theorem provers – everyday we are faced with making decisions based on uncertain and incomplete information
bird(X) flies(X)bird(tweety)
flies(tweety)
By modus tollens: flies(tweety) bird(tweety)
Which is incorrect, the rule, premise 1, or premise 2?
p358
Circumscription
Circumscription can be viewed as a more powerful version of the closed-world assumption
Predicates are introduced to denote the “abnormality” of objects in obeying particular rules
Eg. Birds fly can be written as
bird(X) abnormal1(X) flies(X)
The predicate abnormal1(X) is said to be circumscribed. abnormal1(X) may be assumed unless abnormal1(X) is explicitly known.
p358
Model Preference
Introduces the concept of preferred models, where a sentence is entailed from a KB if it is true in all preferred models (rather than in all models)
One model is preferred to another if it has fewer abnormal objects
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Model Preference - Example
Nixon’s diamond:
republican(nixon) quaker(nixon)
republican(X) abnormal2(X) pacifist(X)
quaker(X) abnormal3(X) pacifist(X)
Was Nixon a pacifist? With both abnormals circumscribed there are two
preferred models, each with one abnormal, hence no definitive conclusion. Only if we were to assert that religious beliefs take precedence over political ones could a conclusion be drawn: prioritised circumscription
p359
Default logic
Default logic is a formalism using default rules to generate non-monotonic conclusions
A default rule looks like this:
P: J1,…,Jn / C
Given a prerequisite P, some justifications Ji and a conclusion C
If any of the justifications are proven false, then the conclusion may not be drawn, eg:
bird(X) : flies(X) / flies(X)
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Extending default rules
The extension of a default theory is the maximal set of consequences of the theory
That is, an extension S consists of the original known facts and a set of drawn conclusions, such that
1) no additional conclusions can be drawn
2) the justification of every default conclusion is consistent with S
Returning to Nixon’s diamond, the both pacifist and pacifist are extensions. As before, prioritised schemes allow some rules to be preferred to others
p359
Semantic networks
Semantic networks are a graphical representation for representing knowledge
Long used in philosophy before computer science
This network by Porphyry
C. 300BC
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Semantic networks cont.
There are many different kinds of semantic network Common to all is that they are a declarative graphic
representation of knowledge, or to support reasoning about knowledge
Some networks are defined to be necessarily true, given their true definitions
Others are used to model human reasoning and can handle uncertainty
p350
Semantic networks – Planet Barklon
“Imagine it is the year 3000 and that a group of space scientists have discovered a planet in a far off galaxy. This planet is called Barklon. Scientists have been observing for 5 years and have built up quite a database – about the peoples, animals and plants on Barklon.
Back at Galactic HQ, your job is to compile a report on the planet Barklon for your superiors. Your superiors are specifically interested in the supplied 14 features. All of the relevant database information is provided with each question.”
Extract from Ford & Billington (2000) “Strategies in Human Non-Monotonic Reasoning”
Planet Barklon Instructions
Twilbers are usually jadds
All jadds are muffers
Kragded is a twilber
Is Kragded a muffer?
The most reasonable answer is B, likely yes.
Barklon Answers
For some questions, difficult to assign “correct” answers – depends on reasoning technique. There are accepted answers though:
1) B 2) B 3) B 4) D 5) E 6) strong E, weak D
7) E 8) E 9) E 10) B 11) D 12) B 13) B 14) E
Graphical representation
All members of the plant species zillo are small
Small plant species are usually found in deserts
Garffi is a zillo
Nagdals are usually spotted plants.
All spotted plants are found in deep caves.
Croldor is a nagdal.
Trendors are usually green animals.
Green animals are usually found on cliffs.
Stordy is a trendor.
Strong vs Weak Specificity
Consider problem 6Zugs are usually not vlogs.
Zugs are usually striped creatures.
Striped creatures are usually vlogs.
Duss is a zug.
Depending on the reasoning method, two possible answers, “likely no” or “can’t tell”
Should two “usually” links be treated equivalently as a single link?
Strong specificity allows transitivity of “usually” Saying “Zugs are usually striped are usually vlogs” is
equivalent to saying “Zugs are usually vlogs” Under this interpretation, the network cannot be
decided and the correct answer is E
Strong Specificity
Weak specificity allows transitivity of “usually” but weaken successive uses
Saying “Zugs are usually striped are usually vlogs” is weaker than saying “Zugs are usually vlogs”
When we then explicitly say that “Zugs are not usually vlogs”, this is the stronger statement
Hence the answer is D
Weak Specificity
Very important to be able to reason about time Representing explicit points is often problematic –
requires either real-valued variables, or quantise time into approriate discrete steps
Using intervals then provides a way to qualitatively reason about time, abstracting away the specific quantified points
Very useful in planning and scheduling domains
Temporal Reasoning
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In computer science, attributed to Allen (1983), but can be dated at least back to Broad (1945)
Between any two intervals there are 13 possible relations
I = <before, meets, overlaps, starts, during, finishes, equals>
The first 6 have inverses to make up to 13.
Interval Algebra
p338
Interval Algebra
Interval Algebra
Determining consistency within temporal constraint networks is difficult because possible relations between intervals are disjunctions – so this or this or this or … can hold.
Can and has been translated into SAT There are a number of tractable subclasses of TR, just
as there are for SAT
Interval Algebra