knowledge representation
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Rushdi Shams, Dept of CSE, KUET, Bangladesh 1
Knowledge Representation IIKnowledge Representation IILogicsLogics
Artificial IntelligenceArtificial IntelligenceVersion 1.0Version 1.0
There are 10 types of people in this world- who understand binary There are 10 types of people in this world- who understand binary and who do not understand binaryand who do not understand binary
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Propositional LogicPropositional Logic
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IntroductionNeed formal notation to represent
knowledge, allowing automated inference and problem solving.
One popular choice is use of logic.Propositional logic is the simplest.
Symbols represent facts: P, Q, etc..These are joined by logical connectives (and,
or, implication) e.g., P Λ Q; Q RGiven some statements in the logic we can
deduce new facts (e.g., from above deduce R)
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Syntactic Properties of Propositional Logic
If S is a sentence, S is a sentence (negation)If S1 and S2 are sentences, S1 S2 is a sentence
(conjunction)If S1 and S2 are sentences, S1 S2 is a sentence
(disjunction)If S1 and S2 are sentences, S1 S2 is a sentence
(implication)If S1 and S2 are sentences, S1 S2 is a
sentence (bi-conditional)
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Semantic Properties of Propositional Logic
S is true iff S is false
S1 S2 is true iff S1 is true and S2 is true
S1 S2 is true iff S1is true or S2 is true
S1 S2 is true iff S1 is false or S2 is true
i.e., is false iff S1 is true and S2 is false
S1 S2 is true iff S1S2 is true and
S2S1 is true
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Truth Table for Connectives
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Model of a FormulaIf the value of the formula X holds 1 for the
assignment A, then the assignment A is called model for formula X.
That means, all assignments for which the formula X is true are models of it.
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Model of a Formula
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Model of a Formula: Can you do it?
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Satisfiable FormulasIf there exist at least one model of a formula
then the formula is called satisfiable.The value of the formula is true for at least
one assignment. It plays no rule how many models the formula has.
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Satisfiable Formulas
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Valid FormulasA formula is called valid (or tautology) if all
assignments are models of this formula.The value of the formula is true for all
assignments. If a tautology is part of a more complex formula then you could replace it by the value 1.
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Valid Formulas
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Unsatisfiable FormulasA formula is unsatisfiable if none of its
assignment is true in no models
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Logical equivalenceTwo sentences are logically equivalent iff true in same
models: α ≡ ß iff α╞ β and β╞ α
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Deduction: Rule of Inference
1.Either cat fur was found at the scene of the crime, or dog fur was found at the scene of the crime. (Premise)
C v D
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Deduction: Rule of Inference
2.If dog fur was found at the scene of the crime, then officer Thompson had an allergy attack. (Premise)
D → A
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Deduction: Rule of Inference
3.If cat fur was found at the scene of the crime, then Macavity is responsible for the crime. (Premise)
C → M
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Deduction: Rule of Inference
4.Officer Thompson did not have an allergy attack. (Premise)
¬ A
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Deduction: Rule of Inference
5.Dog fur was not found at the scene of the crime. (Follows from 2 D → A and 4. ¬ A). When is ¬ A true? When A is false- right? Now, take a look at the implication truth table. Find what is the value of D when A is false and D → A is true
¬ D
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Rules for Inference: Modus TollensIf given α → β
and we know ¬β Then ¬α
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Deduction: Rule of Inference
6.Cat fur was found at the scene of the crime. (Follows from 1 C v D and 5 ¬ D). When is ¬ D true? When D is false- right? Now, take a look at the OR truth table. Find what is the value of C when D is false and C V D is true
C
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Rules for Inference: Disjunctive SyllogismIf given α v β
and we know ¬α then β
If given α v βand we know ¬β then α
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Deduction: Rule of Inference
7.Macavity is responsible for the crime. (Conclusion. Follows from 3 C → M and 6 C). When is C → M true given that C is true? Take a look at the Implication truth table.
M
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Rules for Inference: Modus PonensIf given α → β
and we know α Then β
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Conjunctive Normal Form (CNF)A formula is in conjunctive normal form
(CNF) if it is a conjunction (AND) of clauses, where a clause is a disjunction (OR) of literals or a single literal.
It is similar to the canonical product of sums form used in circuit theory
All of the following formulas are in CNF:
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Conjunctive Normal Form (CNF)The following formulae are not in CNF:
The above three formulas are respectively equivalent to the following three formulas that are in conjunctive normal form:
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Conjunctive Normal Form (CNF)Eliminate implication with its equivalence.
This will turn P → Q into ¬ P V QUse de Morgan's law to move the ¬ symbol
onto atoms (not sentences), replace:
Perform the following operation:
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CNF
(p ^ ~q) V (r V s) ^ (r V t)(p V r V s ) ^ (p V r V t) ^ (~q V r V s)^(~q V r V t)
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Horn Clause A Horn clause is a clause with at most one positive
literal. Any Horn clause therefore belongs to one of four
categories: 1. A rule: 1 positive literal, at least 1 negative literal.
A rule has the form "~P1 V ~P2 V ... V ~Pk V Q".
2. A fact or unit: 1 positive literal, 0 negative literals. 3. A negated goal : 0 positive literals, at least 1
negative literal. 4. The null clause: 0 positive and 0 negative literals.
Appears only as the end of a resolution proof.
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First Order Logic orFirst Order Logic orFirst Order Predicate Logic orFirst Order Predicate Logic or
Predicate LogicPredicate Logic
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IntroductionPropositional logic is declarativePropositional logic allows partial/disjunctive/negated
information(unlike most data structures and databases)
Meaning in propositional logic is context-independent(unlike natural language, where meaning depends on context)
Propositional logic has very limited expressive power(unlike natural language)E.g., cannot say “if any student sits an exam they either pass or fail”.
Propositional logic is compositional(meaning of B ^ P is derived from meaning of B and of P)
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Introduction You see that we can convert the sentences
into propositional logic but it is difficultThus, we will use the foundation of
propositional logic and build a more expressive logic
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IntroductionWhereas propositional logic assumes the
world contains facts,first-order logic (like natural language)
assumes the world containsObjects: people, houses, numbers, colors,
baseball games, wars, …Relations: red, round, prime, brother of, bigger
than, part of, comes between, …Functions: father of, best friend, one more
than, plus, …
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Syntax of FOL: Basic ElementsConstants KingJohn, 2, NUS,... Predicates Brother, >,...Functions Sqrt, LeftLegOf,...Variables x, y, a, b,...Connectives , , , , Equality = Quantifiers ,
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ExamplesKing John and Richard the Lion heart are
brothersBrother(KingJohn,RichardTheLionheart)
The length of left leg of Richard is greater than the length of left leg of King John> (Length(LeftLegOf(Richard)),Length(LeftLegOf(KingJohn)))
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Atomic Sentences
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Atomic Sentences
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Complex SentencesComplex sentences are made from atomic sentences
using connectives:S, S1 S2, S1 S2, S1 S2, S1 S2,
Example
Sibling(KingJohn,Richard) Sibling(Richard,KingJohn)
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Complex Sentences
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FOL illustrated Five objects-1. Richard the Lionheart2. Evil King John3. Left leg of Richard4. Left leg of John5. The crown
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FOL illustrated Objects are related with
Relations For example, King John
and Richard are related with Brother relationship
This relationship can be denoted by(Richard,John),(John,Richard)
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FOL illustrated Again, the crown and
King John are related with OnHead Relationship-OnHead (Crown,John)
Brother and OnHead are binary relations as they relate couple of objects.
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FOL illustrated Properties are relations
that are unary. In this case, Person can
be such property acting upon both Richard and JohnPerson (Richard)Person (John)
Again, king can be acted only upon JohnKing (John)
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FOL illustrated Certain relationships
are best performed when expressed as functions.
Means one object is related with exactly one object.Richard -> Richard’s left legJohn -> John’s left leg
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Universal quantification<variables> <sentence>
Everyone studies at KUET is smart:
x Studies (x,KUET) Smart (x)
x P is true in a model m iff P is true with x being each possible object in the model
Roughly speaking, equivalent to the conjunction of instantiations of P
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Universal quantification Remember, we had five
objects, let us replace them with a variable x-
1. x ―›Richard the Lionheart
2. x ―› Evil King John3. x ―› Left leg of Richard4. x ―› Left leg of John5. x ―› The crown
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Universal quantification Now, for the quantified
sentencex King (x) Person (x)
Richard is king Richard is Person
John is king John is personRichard’s left leg is king
Richard’s left leg is personJohn’s left leg is king John’s
left leg is personThe crown is king the crown is
person
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Universal quantificationRichard is king Richard is
PersonJohn is king John is personRichard’s left leg is king
Richard’s left leg is person
John’s left leg is king John’s left leg is person
The crown is king the crown is person
Only the second sentence is correct, the rest is incorrect
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A common mistake to avoidTypically, is the main connective with
Common mistake: using as the main connective with :
x Studies (x,KUET) Smart (x)
means “Everyone Studies at KUET and everyone is smart”
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Existential Quantification<variables> <sentence>
Someone studies at KUET is smart:x Studies (x,KUET) Smart (x)
x P is true in a model m iff P is true with x being some possible object in the model
Roughly speaking, equivalent to the disjunction of instantiations of P
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Another common mistake to avoidTypically, is the main connective with
Common mistake: using as the main connective with :
x Studies (x,KUET) Smart (x)
means some guys, if they study in KUET, then they are smart
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Properties of quantifiers
x y is the same as y x
x y is the same as y x
x y is not the same as y x
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Properties of quantifiersx y Loves(x,y)
“There is a person who loves everyone in the world”
y x Loves(x,y)
“Everyone in the world is loved by at least one person”
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Properties of quantifiersQuantifier duality: each can be expressed using
the other
x Likes(x,IceCream) is equivalent to x Likes(x,IceCream)
x Likes(x,Broccoli) is equivalent tox Likes(x,Broccoli)
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Properties of quantifiers Equivalences-1. x P is equivalent to x P2. x P is equivalent to x P3. x P is equivalent to x P4. x P is equivalent to x P
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Equalityterm1 = term2 is true under a given interpretation if
and only if term1 and term2 refer to the same object
E.g., definition of Sibling in terms of Parent:x,y Sibling(x,y) [(x = y) m,f (m = f) Parent(m,x)
Parent(f,x) Parent(m,y) Parent(f,y)]
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Example knowledge baseThe law says that it is a crime for an
American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American.
Prove that Col. West is a criminal
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Example knowledge base... it is a crime for an American to sell weapons to hostile nations:
American(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x)Nono … has some missiles,
Owns(Nono,x)Missile(x)
… all of its missiles were sold to it by Colonel WestMissile(x) Owns(Nono,x) Sells(West,x,Nono)
Missiles are weapons:Missile(x) Weapon(x)
An enemy of America counts as "hostile“:Enemy(x,America) Hostile(x)
West, who is American …American(West)
The country Nono, an enemy of America …Enemy(Nono,America)
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Forward ChainingAmerican(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(x,America) Hostile(x)American(West)Enemy(Nono,America)
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Forward ChainingAmerican(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(x,America) Hostile(x)American(West)Enemy(Nono,America)
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Forward ChainingAmerican(West) Weapon(y) Sells(x,y,z)
Hostile(z) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(x,America) Hostile(x)American(West)Enemy(Nono,America)
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Forward ChainingAmerican(West) Weapon(y) Sells(West,y,z)
Hostile(z) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(x,America) Hostile(x)American(West)Enemy(Nono,America)
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Forward ChainingAmerican(West) Weapon(y) Sells(West,y,z)
Hostile(z) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(x,America) Hostile(x)American(West)Enemy(Nono,America)
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Forward ChainingAmerican(West) Weapon(y) Sells(West,y,z)
Hostile(z) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(x)American(West)Enemy(Nono,America)
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Forward ChainingAmerican(West) Weapon(y) Sells(West,y,z)
Hostile(z) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(Nono)American(West)Enemy(Nono,America)
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Forward ChainingAmerican(West) Weapon(y) Sells(West,y,z)
Hostile(Nono) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(Nono)American(West)Enemy(Nono,America)
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Backward ChainingAmerican(West) Weapon(y) Sells(West,y,z)
Hostile(Nono) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(Nono)American(West)Enemy(Nono,America)
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Backward ChainingAmerican(West) Weapon(y) Sells(West,y,z)
Hostile(Nono) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(Nono)American(West)Enemy(Nono,America)
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Backward ChainingAmerican(West) Weapon(y) Sells(West,y,z)
Hostile(Nono) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(Nono)American(West)Enemy(Nono,America)
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Backward ChainingAmerican(West) Weapon(y) Sells(West,y,z)
Hostile(Nono) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(Nono)American(West)Enemy(Nono,America)
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Backward ChainingAmerican(West) Weapon(y) Sells(West,y,z)
Hostile(Nono) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(Nono)American(West)Enemy(Nono,America)
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Backward ChainingAmerican(West) Weapon(y) Sells(West,y,Nono)
Hostile(Nono) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(Nono)American(West)Enemy(Nono,America)
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Backward ChainingAmerican(West) Weapon(y) Sells(West,y,Nono)
Hostile(Nono) Criminal(x)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(Nono)American(West)Enemy(Nono,America)
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…& the InferenceAmerican(West) Weapon(y) Sells(West,y,Nono)
Hostile(Nono) Criminal(West)
Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(Nono)American(West)Enemy(Nono,America)
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Probability: Logic for Probability: Logic for UncertaintyUncertainty
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Conditional ProbabilityDefinition of conditional probability:
P(a | b) = P(a b) / P(b) if P(b) > 0Product rule gives an alternative formulation:
P(a b) = P(a | b) P(b) = P(b | a) P(a)
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Inference with Probability
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Inference in ProbabilityP(toothache) =
0.108 + 0.012 + 0.016 + 0.064 = 0.2
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Inference in ProbabilityP(cavity V toothache) =
0.108 + 0.012 + 0.072 + .008 + 0.016 + 0.064 = 0.28
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Inference in ProbabilityCan also compute conditional probabilities:
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Inference in ProbabilityCan also compute conditional probabilities:
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Baye’s RuleProduct rule gives an alternative formulation:
P(a b) = P(a | b) P(b) = P(b | a) P(a)
Joining them together, we can find-P(a | b) = P(b | a) P(a)
P(b)
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Application of Bayes’ RuleA doctor knows that the disease meningitis causes the patient to have a stiff neck is 50%Means probability of stiff neck given the probability of having meningitis P(s | m) = 0.5He also knows that in every 50000 patients, 1 may have meningitisMeans probability that a patient has meningitisP (m) = 1/50000He also knows that in every 20 patients, 1 may have stiff neckMeans probability that a patient has meningitisP (m) = 1/20Then, from Bayes’ ruleP(m | s) = P(s | m) P(m)
P(s)
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Application of Bayes’ RuleP(m | s) = P(s | m) P(m)
P(s)= 0.5 X (1/50000)
1/20= 0.0002
Means he can expect only 1 in 5000 patients with a stiff neck
to have meningitis
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ReferencesArtificial Intelligence: A Modern Approach
(2nd Edition)by Russell and NorvigChapter 7, 8, 9, 13
http://www.iep.utm.edu/p/prop-log.htm#H5 http://www.cs.yale.edu/homes/cc392/node5.h
tml
http://www.cs.nyu.edu/courses/spring03/G22.2560-001/horn.html
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Acknowledgement Dr. Adel Elsayed
Research Leader, M3C Lab, University of Bolton, UK
Weiqiang WeiPhD Student, University of Bolton, UK