Applied Artificial Intelligence Unit 2 Semester 3 MSc IT Part 2 Mumbai University

Applied Artificial Intelligence
UNIT-2

Topics To Be Covered
Probability Theory: Joint probability,
conditional probability, Bayes’ theorem,
probabilities in rules and facts of rule
system, cumulative probabilities, rule
system and Bayesian method
Fuzzy Sets and Fuzzy Logic: Fuzzy Sets,
Fuzzy set operations, Types of Member
Functions, Multivalued Logic, Fuzzy Logic,
Linguistic variables and Hedges, Fuzzy
propositions, inference rules for fuzzy
propositions, fuzzy systems, possibility
theory and other enhancement to Logic
PPT BY : MADHAV MISHRA 2

Probability Theory
 In order to discuss probability theory in detail, let us review a few basic concepts of this theory.
 The term probability is defined as a way of turning an opinion or an expectation into a number lying
between 0 and 1.
 It basically reflects the likelihood of an event, or a chance that a particular event will occur.
 Assume a set S(known as sample space) consisting of independent events representing all possible
outcomes of a random experiment.
 Every, non-empty subset A of sample space S is called an event
 The empty set is called an impossible event, whereas S is called a sure event.
 Let us denote the probability of an event A by P(A), it is defined as follows:
P(A) = (No.of outcomesfavourite to A) / (Totalnumberof possible outcomes)
PPT BY : MADHAV MISHRA
3

 The Probability of all events, {A1, A2, ….An}, must sum up to certainty, that is P(A1)+P(A2)…..+P(An)= 1.
 An impossible event has a probability of 0, while a certain event has a probability 1.
 Consider an example of tossing a coin, the probability of throwing two successive heads with a fair coin
is found to be 0.25.
 Let us see how we got this, there are only four possible outcome {HH, HT, TH, TT} of throwing a fair
coin twice successively.
 Here, H represents head, while T represents tail. Since there is only one way of getting HH out of
four, the probability =1/4 = 0.25.
 Axioms of Probability:
 If S represents a sample space and A and B represent events, then the following axioms hold true.
 Here, A’ represents complement of set A.

Joint Probability
 Joint probability is defined as the probability of occurrence of two independent events in conjunction.
 That is, joint probability refers to the probability of both events occurring together.
 The joint probability of A and B is written as P(A ∩ B) or P(A and B). It may be defined as given below:
P(AandB) = P(A)* P(B)
 Two events are said to be independent if the occurrence of one event does not affect the probabilities of
occurrence of the other.
 Consider an example of tossing of two fair coins separately. The probability of getting a head H on tossing the
first coin is denoted by P(A) = 0.5, and the probability of getting a head on tossing the second coin is denoted
by P(B) = 0.5. The probability of getting H on both the coins is called Joint Probability and is represented as P(A
and B). It is calculated as follows:
P(A and B) = P(A) * P(B)
= 0.5 * 0.5
= 0.25
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 Similarly, the probability of getting a head H on tossing one or both coins can be calculated. It is called
union of the probabilities P(A) and P(B), and is denoted by P(A U B), it is also written as P(A or B).
• It can be calculated for the above example as follows:
P(A or B) = P(A) + P(B) - P(A) * P(B)
= 0.5 + 0.5 – 0.25
= 0.75

Bayes’ Rule

Bayes’ theorem
 Bayes’ theorem was developed by
mathematician Thomas Bayes in the
year 1763.
 This theorem provides a
mathematical model for reasoning
where prior belief’s are combined
with evidence to get estimates of
uncertainty.
 It relates the conditional probability
and probabilities of events.

Extension of Bayes’ Theorem
 So far we have studied Bayes’ theorem with one hypothesis and one event or evidence.
 We can extend Bayes’ theorem to include more than two events.
 Different possibilities are discussed as given below:
 One Hypothesis & Two Evidence
 One Hypothesis & Multiple Evidence
 Chain Evidence
 Multiple Hypothesis & Single Evidence
 One Hypothesis & Multiple Evidence

Probabilities in rules and facts of rule based system
 Basic probability theory can be used when we have probable facts and rules.
 In real-life situations, facts may also be probably true.
 Probabilities can be added to both facts and rules.
 It should be noted that probabilities are specified by domain expert on the basis of their experience.
 The following are some examples to illustrate the incorporation of probabilities in facts and rules represented in
PROLOG:
 A fact ‘battery in a randomly picked computer is dead 2% of the time’ can be expressed in PROLOG as
battery_dead_computer(0.02)
 This fact indicates that battery is dead in computer is sure with probability 0.02.
 Similarly probabilities can also be added to rules.
 Consider the following probable rules and their corresponding PROLOG representation
 The rule ‘ IF 25% of the time when the computer does not work, it is true that it’s battery is dead’ can be
expressed in PROLOG as battery_dead_computer(0.25): computer_not_start(1.0)
 Here, 0.25 is the rule probability.
16

Cumulative Probabilities
 For the rules discussed in the above section, if we wish to reason about whether the battery in
computer is dead, we should gather all relevant rules and facts.
 It is very important to combine the probabilities from the facts and successful rules to get a
cumulative probability of the battery being dead.
 The following two situations will arise:
 If sub goals of a rule are probable, then the probability of the rule of succeed should take
care of the probable sub goals.
 If rules with the same conclusion have different probabilities, then the overall probability of
the rule has to be found.

 The first situation is resolved by simple computing cumulative probability of the conclusion with the
help of and- combination assuming that all sub goals are independent.
 In this case, probabilities of sub goals in the right side of rule of rule are multiplied using joint
probability formula as shown below:
Prob(A and B and C and…) = Prob(A) * Prob(B) * Prob(C) * …
 The second situation is handled by using or-combination to get the overall probability of
predicate in the head of rule.
 If events are mutually independent, the following formula is used to obtain the OR Probability:
Prob(A and B and C and…) = 1 – [(1- Prob(A)) (1- Prob(B)) (1- Prob(C))….]

Rule based system
using probability
 Let us develop a simple rule-
based system for the diagnosis
of malfunctioning of some
equipment, say a landline
telephone using the concept of
probabilities.
 As a first step, various situations
under which the telephone does
not work properly are identified
with the help of experts of this
domain.
 For Example, we can think of the
following reasons for the
malfunctioning of telephones:
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Bayesian Method
 Bayesian network are useful for both inferential exploration of previously undetermined
relationships among variables and descriptions of these relationship upon discovery.
 The following are some advantages and disadvantages of Bayesian method:
 Advantages:
 Bayesian method is based on as strong theoretical foundation in probability theory, it is
currently the most advanced of all certainty reasoning methods.
 This method has well-defined semantics for decision making.
 Disadvantages:
 The system using Bayesian approach needs quite a large amount of probability data to
construct a knowledge base.
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 Bayesian theory provides an attractive basis for an uncertain reasoning system.
 Several mechanism have been developed to utilize the power of this theory and the ease of its
implementation.
 Some of these are listed below:
 Bayesian Belief Network.
 Certainty Factor Theory.
 Dempster- Shafer Theory.

Fuzzy Sets
 The concept of a fuzzy set was published in 1965 by Lotfi A. Zadeh (see also Zadeh 1965).
 Since that seminal publication, the fuzzy set theory is widely studied and extended.
 Its application to the control theory became successful and revolutionary especially in seventies
and eighties, the applications to data analysis, artificial intelligence, and computational
intelligence are intensively developed, especially, since nineties.
 The theory is also extended and generalized by means of the theories of triangular
norms and aggregation operators.

 Fuzzy sets can be considered as an extension and gross
oversimplification of classical sets.
 It can be best understood in the context of set membership.
 Basically it allows partial membership which means that it
contain elements that have varying degrees of membership in
the set.
 From this, we can understand the difference between classical
set and fuzzy set.
 Classical set contains elements that satisfy precise properties
of membership while fuzzy set contains elements that satisfy
imprecise properties of membership.

Classical Set

Fuzzy Set
Operations

Types of Membership Functions
35

Multivalued Logic
 Multivalued logic is defined as a logical system in which there are more than two truth values.
 Traditionally, two-valued logic is binary logic where there are only two possible truth values, i.e.
Truth & False for any proposition.
 An obvious extension of classical two-valued logic is three-valued logic denoting truth, false &
indeterminate (indeterminacy) represented as (1, 0 and 1/2).
 Three- valued logic can be easily generalized to n-valued logic.
 Relations used in Multi-valued logic are:
 Relation of multi-valued logic to classical logic.
 Relation of multi-valued logic to fuzzy logic.

Fuzzy Logic

Linguistic
Variables &
Hedges

Inference Rules For Fuzzy Propositions
Fuzzy Inference
 Fuzzy inference is the process of obtaining new knowledge through existing knowledge.
 Knowledge is most commonly represent in the form of rules or proposition for example
 “if x is A then y is B” (Where A and B are linguistic values defined by fuzzy sets on universes of
discourse X and Y).
 A rule is also called a fuzzy implication.
 “x is A” is called the antecedent or premise and “y is B” is called the consequence or conclusion.
 The two important inferring processes are –
 Generalized modus Ponens (GMP)
 Generalized modus Tollens (GMT)

Inference rules:
 Inference rules are the templates for generating valid arguments.
 Inference rules are applied to derive proofs in artificial intelligence, and the proof is a sequence of the
conclusion that leads to the desired goal.
 In inference rules, the implication among all the connectives plays an important role.
 Following are some terminologies related to inference rules:
 Implication: It is one of the logical connectives which can be represented as P → Q. It is a Boolean
expression.
 Converse: The converse of implication, which means the right-hand side proposition goes to the left-
hand side and vice-versa. It can be written as Q → P.
 Contrapositive: The negation of converse is termed as contrapositive, and it can be represented as ¬ Q
→ ¬ P.
 Inverse: The negation of implication is called inverse. It can be represented as ¬ P → ¬ Q.

From the above term some of the compound statements are equivalent to each other, which we can prove
using truth table:
Hence from the above truth table, we can prove that P → Q is equivalent to ¬ Q → ¬ P, and Q→ P is
equivalent to ¬ P → ¬ Q.

85
Types of Inference rules:
1. Modus Ponens:
The Modus Ponens rule is one of the most important rules of inference, and it states that if P and P → Q
is true, then we can infer that Q will be true. It can be represented as:
Example:
Statement-1: "If I am sleepy then I go to bed" ==> P→ Q
Statement-2: "I am sleepy" ==> P
Conclusion: "I go to bed." ==> Q.
Hence, we can say that, if P→ Q is true and P is true then Q will be true.
Proof by Truth table:

2. Modus Tollens:
The Modus Tollens rule state that if P→ Q is true and ¬ Q is true, then ¬ P will also true. It can be
represented as:
Statement-1: "If I am sleepy then I go to bed" ==> P→ Q
Statement-2: "I do not go to the bed."==> ~Q
Statement-3: Which infers that "I am not sleepy" => ~P
Proof by Truth table:

3. Disjunctive Syllogism:
4. Addition:
5. Simplification:
6. Resolution:

Inference rules:
 Inference rules are the templates for generating valid arguments.
 Inference rules are applied to derive proofs in artificial intelligence, and the proof is a sequence of the
conclusion that leads to the desired goal.
 In inference rules, the implication among all the connectives plays an important role.
 Following are some terminologies related to inference rules:
 Implication: It is one of the logical connectives which can be represented as P → Q. It is a Boolean
expression.
 Converse: The converse of implication, which means the right-hand side proposition goes to the left-
hand side and vice-versa. It can be written as Q → P.
 Contrapositive: The negation of converse is termed as contrapositive, and it can be represented as ¬ Q
→ ¬ P.
 Inverse: The negation of implication is called inverse. It can be represented as ¬ P → ¬ Q.

Fuzzy Systems
CoveredBefore In FuzzyLogic TopicBefore …
Let us Reviseit Again….!

Possibility Theory
 Possibility theory was introduced in the year 1978.
 Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an
alternative to probability theory.
 A possibility measure can be seen as reasonable measure in Dempster-Shafer theory of
evidence.
 Possibility theory can be seen as an upper probability.(any possibility distribution that defines a
unique set of probability distribution)

Other
Enhancement
To Logic

The End

Applied Artificial Intelligence Unit 2 Semester 3 MSc IT Part 2 Mumbai University

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