Unit 6 Uncertainty.pptx

Sanjivani Rural Education Society’s
Sanjivani College of Engineering, Kopargaon-423603
(AnAutonomous Institute Affiliated to Savitribai Phule Pune University, Pune)
NAAC ‘A’GradeAccredited, ISO 9001:2015 Certified
Department of Information Technology
(NBA Accredited)
Department:- Information Technology
Name of Subject:- Artificial Intelligence
Class:- TYIT
Subject Code:- IT313
Sanjivani College of Engineering, Kopargaon Dept of Information Technology

Course Objectives:
1. To understand the basic principles of Artificial Intelligence
2. To provide an understanding of uninformed search strategies.
3. To provide an understanding of informed search strategies.
4. To study the concepts of Knowledge based system.
5. To learn and understand use of fuzzy logic and neural networks.
6. To learn and understand various application domain of Artificial Intelligence.

Uncertainty
Till now, we have learned knowledge representation using first-order logic and
propositional logic with certainty, which means we were sure about the predicates.
With this knowledge representation, we might write A→B, which means if A is true then B
is true, but consider a situation where we are not sure about whether A is true or not then
we cannot express this statement, this situation is called uncertainty.
 So to represent uncertain knowledge, where we are not sure about the predicates, we need
uncertain reasoning or probabilistic reasoning.

Non Monotonic Reasoning
 In Non-monotonic reasoning, some conclusions may be invalidated if we add some more
information to our knowledge base.
 Logic will be said as non-monotonic if some conclusions can be invalidated by adding more
knowledge into our knowledge base.
 Non-monotonic reasoning deals with incomplete and uncertain models.
 "Human perceptions for various things in daily life”, is a general example of non-monotonic
reasoning.
 Example: Let suppose the knowledge base contains the following knowledge;
Birds can fly
Penguins cannot fly
Pitty is a bird
So from the above sentences, we can conclude that Pitty can fly.
However, if we add one another sentence into knowledge base "Pitty is a penguin", which concludes
"Pitty cannot fly", so it invalidates the above conclusion.

Non Monotonic Reasoning
Advantages of Non-monotonic reasoning:
 For real-world systems such as Robot navigation, we can use non-monotonic reasoning.
 In Non-monotonic reasoning, we can choose probabilistic facts or can make assumptions.
Disadvantages of Non-monotonic Reasoning:
 In non-monotonic reasoning, the old facts may be invalidated by adding new sentences.
 It cannot be used for theorem proving.

Logics for Non Monotonic Reasoning
Suppose the purchasing agent is investigating purchasing holidays. A resort may be adjacent to a
beach or away from a beach. This is not symmetric; if the resort was adjacent to a beach, the
knowledge provider would specify this.
 Thus, it is reasonable to have the clause away_from_beach ← ∼ on_beach. This clause enables an
agent to infer that a resort is away from the beach if the agent is not told it is adjacent to a beach.
A cooperative system tries to not mislead. If we are told the resort is on the beach, we would
expect that resort users would have access to the beach. If they have access to a beach, we would
expect them to be able to swim at the beach. Thus, we would expect the following defaults:
beach_access ←on_beach 𝖠 ∼ abbeach_access.
swim_at_beach ←beach_access 𝖠 ∼ abswim_at_beach.
A cooperative system would tell us if a resort on the beach has no beach access or if there is no
swimming. We could also specify that, if there is an enclosed bay and a big city, then there is no
swimming, by default:
abswim_at_beach ←enclosed_bay 𝖠big_city 𝖠 ∼ abno_swimming_near_city.

Semantic Network
Semantic Networks or Semantic Net is a knowledge representation technique used for
propositional information.
Though its versions were long being used in philosophy, cognitive science (in the form of
semantic memory), and linguistics, Semantic Network’s implementation in computer
science was first developed for artificial intelligence and machine learning.
 It is a knowledge base that represents concepts in a network and the systematic relations
between them.
Semantic Network is a directed or undirected graph consisting of vertices. These vertices
represent concepts and edges, which further represent semantic relations between
concepts, mapping or connecting semantic fields.
 Moreover, it is termed as Associative Networks, as it processes the knowledge about
accepted meanings in adjacent regions.

Semantic Network
A semantic net (or semantic network) is
a knowledge representation technique used for
propositional information.
So it is also called a propositional net.
Semantic nets convey meaning. They are two
dimensional representations of knowledge.
 Mathematically a semantic net can be
defined as a labeled directed graph.
The main idea behind semantic nets is that
the meaning of a concept comes from the ways
in which it is connected to other concepts.
In semantic net, information is represented
as a set of nodes connected to each other by
set of labeled arcs, which represent
relationships among the nodes.
INHERITANCE REASONING
 Semantic net allows us to perform inheritance reasoning as all
members of a class will inherit all the properties of super class .
 We could use inheritance to derive the additional relation.
Semantic nets allow multiple inheritance.
 So an object can belong to more than one category and a
category can be a subset of more than one another category.

Semantic Network
SEMANTIC NETWORK EXAMPLES:
Some of the examples of Semantic Networks are:
1. WordNet: A lexical database of English that groups English words into sets of synonyms
(synsets), provides definitions, and records semantic relations between them.
2. Gellish Model: It is a formal language that is defined as a network of relations between
concepts and names of concepts. It started out as an engineering modeling language
("Generic Engineering Language", giving it the name, "Gellish")
3. Logical Descriptions: Semantic Networks can also be used to represent logical
descriptions like Charles Sanders Peirce’s existential graphs or John F. Sowa's conceptual
graphs.

Semantic Network
 AI agents have to store and organize information in their memory. One of the ways they do
this is by using semantic networks.
 Semantic networks are a way of representing relationships between objects and ideas.
For example, a network might tell a computer the relationship between different animals (a
cat IS A mammal, a cat HAS whiskers). Below is an example image of a semantic network;

Components of Semantic Network
Semantic Networks can further be defined by specifying its fundamental components, which are:
1. Lexical Components - Consists of;
 Nodes represent the object or concept.
 Links: Denoting relation between nodes.
 Labels: Denoting particular objects & relations.
2. Structural Component - Here the links and nodes form a directed graph wherein the labels are
placed on the link and nodes.
3. Semantic Component - The meanings here are related to the links and labels of nodes, whereas
the facts are dependent on the approved areas.
4. Procedural Part - The creation of new links and nodes is permitted by constructors, whereas the
destructors are responsible for the removal of links and nodes.

Example
· Tom caught a cat
· Tom is owned by rashan.
· Tom is brown in colour.
· Dogs like bones.
· The dog sat on the mat.
· A dog is a mammal.
· A cat is an instance animal
· All mammals are animals.
· Mammals have fur.
 This network contains examples of isa relation, as well as some other, more domain-specific
relations like sat_on, like, caught, is_coloured, is_owned_by.
 In this network, we could use inheritance to derive the additional relation;
sat_on (Tom,Mat)

Advantages & Disadvantages
Advantages:
Semantic nets have the ability to
represent default values for categories. In
the above figure John has one leg while he
is a person and all persons have two legs.
So persons have two legs has only
default status which can be overridden by
a specific value.
1. They convey some meaning in a
transparent manner.
2. They nets are simple and easy to
understand.
3. They are easy to translate into PROLOG.
Disadvantages:
semantic
One of the drawbacks of
network is that the links between the
objects represent only binary relations.
For example, the sentence
Run(RajdhaniExpress, Chandigarh, Delhi,
Tomorrow) cannot be asserted directly.
There is no standard definition of link
names.

Truth Maintenance System
 TMS is a “Truth Maintenance System” which
implements to permit a form of non-monotonic
reasoning by permitting the addition of changing
statements to a knowledge base.
 It is also known as “Belief-Revision” or “Revision
Management Systems“.
 Role of Truth Maintenance System:
1.The main job of TMS is to maintain “consistency of
knowledge” being used by the problem solver and not
to perform any inference functions.
2.TMS also gives the inference – component, the
latitude to perform non-monotonic inferences.
3.When discoveries made, this more recent
information can displace previous conclusions that are
no longer valid.
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K. S. Ubale

Justification based Truth Maintenance Systems
 JTMS utilizes three types of nodes, where each node is
associated with an assertion;
1. Premises: Their justifications (provided by the IE) have no
antecedents.
2. Contradictions: They are no different from other nodes
expect that the IE has explicitly designated them as
contradictions. It a contradiction node becomes believed,
JTMS must signal the IE and the IE must assure that the
contradiction node is no longer believed.
3. Assumptions: These are also designated by the IE. An
assumption is enabled if the IE has instructed the JTMS to
believe it. If an assumption node has a valid justification,
then it is treated as a “regular” node.
In dependency networks the following
symbols are used for representation :

JTMS node’s labels
 Nodes can be either IN or OUT where;
1. IN means “believed”.
2. OUT means “not believed”.
N1 is “OUT”
 Consider the following 4 cases;
N1 is “IN”
(not N1) is “IN”
(not N1) is “OUT”
Note: A node being “IN” does not mean a node is “true”.
Contradiction (not N1) is true
N1 is true Unknown

JTMS justifications
Here N1 and N2 are the antecedents of justification, J1, and N3 is the consequent. The
informant is ignored or it may record information from the external systems.
A justification is valid if its antecedents are : IN.
 Example justification:
N1
N2
J1
N3

Prepositional specification of a JTMS
JTMS is defined by the following two sets:
 The set of enabled assumptions A.
 The set of justifications J.
The set of justifications grows monotonically, because justifications cannot be removed.
Justifications are Horn clauses, therefore we can think of them as implications in PL.
 JTMS nodes are propositional symbols. They form the JTMS belief set, Bel.
The fundamental task of JTMS is to answer queries
about whether a given node holds(is “N”) in a given set
of beliefs and justifications. JTMS classified node N, as
“IN” iff;
1. The node is an enabled assumption or a premise.
2. Bel U J |--PL N
Otherwise node N is “ OUT ”.

Example of JTMS: How it helps the IE to improve its efficiency
Consider the following example:
JTMS maintains the records of previous IE work, which is why there is no need to re-compute the label for G in the
third case. It has already been computed in case (a). That is, it is already known that if D and F are IN, G will be IN.
Changing the JTMS state can be caused by;
1. Adding a new assertion(enabled assumption or premise) or a justification.
2. Retracting an enabled assumption.
Whenever the IE changes the JTMS state, the later must;
1. Identify the change exactly.
2. Carry as much work forward as it is logically possible.

Adding information to JTMS
 There are three ways to add information to the JTMS state;
1. Add a justification: Check whether the consequent of the justification is IN. If yes, do nothing. If
no, check if the justification is valid(ie. Its antecedents are IN) and if yes, make the new
justification the supporting justification for the consequent.
2. Enable an assumption:
i. If the node was not an assumption and was IN(for which it must have had a valid justification),
remove its valid support and mark it as an enabled assumption.
ii. Check if this assumption was IN. If yes, do nothing.
iii. Check any justification where this assumption is an antecedent to see if it now becomes valid
and if yes change the status of the consequent.
3. Declare a premise.
In all these cases, the JTMS must;
1. Create a new node or justification, if necessary.
2. Propagate the consequences(run the “Propagate-Inness” algorithm).

Propagating Inness: Example
 Consider the following dependency network;
Let A becomes an enabled assumption, The network
changes as follows;
Note that we only change nodes from OUT to IN. Because there is a finite number of nodes, this process must
terminate at some point.

Propagating Outness: Example
 Consider the following dependency network;
Consider the modified version of the example network;

How JTMS simulates default reasoning
 Consider the following
default rules;
MA  A
MB  B
MC  C
A & B  ┴
B & C  ┴
Assume that A, B and C are all declared as enabled
assumptions. Here is the corresponding network;
Assume the IE decides to retract A.
The resulting network is the following;
To get rid of the new contradiction, assume that the
IE decides to retract B;
Note that the resulting network does not satisfy the requirement that
in a default theory an assumption must be IN unless it causes a
contradiction. To correct this situation, A must be enabled again. The
resulting network is the following;

 This is a simple TMS in that it does not know anything about the structure of the
assertions themselves.
 Each supported belief (assertion) in has a justification.
Each justification has two parts:
An IN-List -- which supports beliefs held.
An OUT-List -- which supports beliefs not held.
 An assertion is connected to its justification by an arrow. One assertion can feed another
justification thus creating the network.
 Assertions may be labelled with a belief status. An assertion is valid if every assertion in
the IN-List is believed and none in the OUT-List are believed.
An assertion is non-monotonic is the OUT-List is not empty or if any assertion in the IN-List
is non-monotonic.

 A Justification-based truth maintenance system (JTMS) is a simple TMS where one can examine the
consequences of the current set of assumptions.
In JTMS labels are attached to arcs from sentence nodes to justification nodes. This label is either
"+" or "-". Then, for a justification node we can talk of its in-list, the list of its inputs with "+" label,
and of its out-list, the list of its inputs with "-" label.
 The meaning of sentences is not known. We can have a node representing a sentence p and one
representing ~p and the two will be totally unrelated, unless relations are established between them
by justifications.
 For example, we can write:
 Which says that if both p and ~p are IN we have a contradiction.
 The association of IN or OUT labels with the nodes in a dependency
network defines an in-out-labeling function.
 This function is consistent if:
1. The label of a justification node is IN iff the labels of all the
sentence nodes in its in-list are all IN and the labels of all the
sentence nodes in its out-list are OUT.
2. The label of a sentence node is IN iff it is a premise, or an enabled
assumption node, or it has an input from a justification node with
label IN.

Example
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Here is a simple dependency network:
This last command
will recognize that a
contradiction exists and will invoke a contradiction
handler that should state that the node
"contradiction" is a contradiction and that some of
the assumptions in {A, C} should be retracted.

Fuzzy logic
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 Fuzzy logic is an approach to variable processing that allows for multiple possible truth values to
be processed through the same variable.
 Fuzzy logic attempts to solve problems with an open, imprecise spectrum of data and heuristics that
makes it possible to obtain an array of accurate conclusions.
 Fuzzy logic is designed to solve problems by considering all available information and making the
best possible decision given the input.
 Fuzzy logic in AI:
•In artificial intelligence (AI) systems, fuzzy logic is used to imitate human reasoning and cognition.
Rather than strictly binary cases of truth, fuzzy logic includes 0 and 1 as extreme cases of truth but
with various intermediate degrees of truth.
• As a result, fuzzy logic is well-suited for the following;
1. Engineering for decisions without clear certainties and uncertainties, or with imprecise data --
such as with natural language processing technologies; and
2. Regulating and controlling machine outputs, according to multiple inputs/input variables -- such as
with temperature control systems.

What is Fuzzy Logic?
 The 'Fuzzy' word means the things that are not clear. Sometimes, we cannot
decide in real life that the given problem or statement is either true or false.
At that time, this concept provides many values between the true and false
and gives the flexibility to find the best solution to that problem.
Example of Fuzzy Logic as comparing to Boolean Logic;
1. Fuzzy logic contains the multiple logical values and these values are the
truth values of a variable or problem between 0 and 1. This concept
provides the possibilities which are not given by computers, but similar to
the range of possibilities generated by humans.
2. In the Boolean system, only two possibilities (0 and 1) exist, where 1 denotes
the absolute truth value and 0 denotes the absolute false value. But in the
fuzzy system, there are multiple possibilities present between the 0 and 1,
which are partially false and partially true.
3. The Fuzzy logic can be implemented in systems such as micro-controllers,
workstation-based or large network-based systems for achieving the
definite output. It can also be implemented in both hardware or software.

Characteristics of Fuzzy Logic
1. This concept is flexible and we can easily understand and implement it.
2. It is used for helping the minimization of the logics created by the human.
3. It is the best method for finding the solution of those problems which are suitable for
approximate or uncertain reasoning.
4. It always offers two values, which denote the two possible solutions for a problem and
statement.
5. It allows users to build or create the functions which are non-linear of arbitrary complexity.
6. In fuzzy logic, everything is a matter of degree.
7. In the Fuzzy logic, any system which is logical can be easily fuzzified.
8. It is based on natural language processing.
9. It is also used by the quantitative analysts for improving their algorithm's execution.
10.It also allows users to integrate with the programming.

Architecture of a Fuzzy Logic System
 In the architecture of the Fuzzy Logic system, each component plays an important role.
 The architecture consists of the different four components which are given below.
1. Rule Base
2. Fuzzification
3. Inference Engine
4. Defuzzification
Following diagram shows the architecture or process of a Fuzzy Logic system:

1. Rule Base:
Rule Base is a component used for storing the set of rules and the If-Then conditions given by the experts
are used for controlling the decision-making systems. There are so many updates that come in the Fuzzy
theory recently, which offers effective methods for designing and tuning of fuzzy controllers. These updates or
developments decreases the number of fuzzy set of rules.
2. Fuzzification:
Fuzzification is a module or component for transforming the system inputs, i.e., it converts the crisp
number into fuzzy steps. The crisp numbers are those inputs which are measured by the sensors and then
fuzzification passed them into the control systems for further processing.
 This component divides the input signals into following five states in any Fuzzy Logic system:
1. Large Positive (LP)
2. Medium Positive (MP)
3. Small (S)
4. Medium Negative (MN)
5. Large negative (LN)

3. Inference Engine:
This component is a main component in any Fuzzy Logic system (FLS), because all the
information is processed in the Inference Engine.
 It allows users to find the matching degree between the current fuzzy input and the rules.
After the matching degree, this system determines which rule is to be added according to the
given input field. When all rules are fired, then they are combined for developing the control
actions.
4. Defuzzification:
 Defuzzification is a module or component, which takes the fuzzy set inputs generated by
the Inference Engine, and then transforms them into a crisp value.
It is the last step in the process of a fuzzy logic system. The crisp value is a type of value which is
acceptable by the user. Various techniques are present to do this, but the user has to select the best
one for reducing the errors.

Membership function
The membership function is a function which represents the
graph of fuzzy sets, and allows users to quantify the linguistic
term. It is a graph which is used for mapping each element of x
to the value between 0 and 1.
This function is also known as indicator or characteristics
function.
This function of Membership was introduced in the first papers
of fuzzy set by Zadeh. For the Fuzzy set B, the membership
function for X is defined as; μB:X → [0,1]. In this function X, each
element of set B is mapped to the value between 0 and 1. This is
called a degree of membership or membership value.
 The fuzzy set approach provides a much better representation
of the tallness of a person..
The membership function defines the fuzzy set for the possible
values underneath of it on the horizontal axis. The vertical axis, on
a scale of 0 to 1

Linguistic variable, linguistic term, fuzzy rules
Linguistic variable: A linguistic variable is a variable whose values are sentences in a natural or
artificial language.
For example, the values of the fuzzy variable height could be tall, very tall, very very tall, somewhat
tall, not very tall, tall but not very tall, quite tall, more or less tall.
 Tall is a linguistic value or primary term
If age is a linguistic variable then its term set is T(age) = { young, not young, very young, not very
young,…… middle aged, not middle aged, … old, not old, very old, more or less old, not very old,…not
very young and not very old,…}.
 Fuzzy rules are useful for modeling human thinking, perception (Opinion,view) and judgment.
A fuzzy if-then rule is of the form “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, respectively. “x is A” is called antecedent and “y is B” is
called consequent.
 Example: If pressure is high, then volume is small.
If the road is slippery, then driving is dangerous.
If the fruit is ripe, then it is soft.

Membership function
 A membership function provides a measure of the degree of similarity of an element
to a fuzzy set.
 Membership functions can;
– either be chosen by the user arbitrarily, based on the user’s experience (MF chosen by
two users could be different depending upon their experiences, perspectives, etc.)
–Or be designed using machine learning methods (e.g., artificial neural networks,
genetic algorithms, etc.)
There are different shapes of membership functions;
1. Triangular,
2. Trapezoidal,
3. Gaussian, etc

Types of Membership function
 Triangular membership function:
A triangular membership function is specified by
three parameters {a, b, c} a, b and c represent the x
coordinates of the three vertices of µA (x) in a fuzzy
set A (a: lower boundary and c: upper boundary
where membership degree is zero, b: the centre
where membership degree is 1)
 Trapezoid membership function:
A trapezoidal membership function is specified by
four parameters {a, b, c, d} as follows;
 Gaussian membership function:
– c: centre
– s: width
– m: fuzzification factor (e.g., m=2)

Classical Set
 Classical Set:
A Set is any well defined collection of objects.
An object in a set is called an element or
member of that set. Sets are defined by a simple
statement, describing whether a particular
element having a certain property belongs to
that particular set. A = {a1 ,a2 ,a3 ,……,an }
 If the elements ai (i = 1,2,3,….,n) of a set A are
subset of universal set X, then set A can be
represented for all elements x ϵ X by its
x ϵ X
characteristics function µA (x) = 1 if
otherwise 0.

Fuzzy Set
 Fuzzy Set:
 Fuzzy sets theory is an extension of classical set theory.
Elements have varying degree of membership. A logic based on two
truth values, True and False is sometimes insufficient when describing
human reasoning.
 Fuzzy Logic uses the whole interval between 0 (false) and 1 (true) to
describe human reasoning.
 A Fuzzy Set is any set that allows its members to have different degree
of membership, called membership function, having interval [0,1].
Fuzzy Logic is derived from fuzzy set theory. Many degree of
membership (between 0 to 1) are allowed. Thus a membership function
µA (x) is associated with a fuzzy sets Ã such that the function maps
every element of universe of discourse X to the interval [0,1]. The
mapping is written as: µÃ (x): X → [0,1].
Fuzzy set is defined as follows; If X is an universe of discourse and x is
a particular element of X, then a fuzzy set A defined on X and can be
written as a collection of ordered pairs A = {(x, µÃ (x)), x є X }
Example:
Let X = {g1 , g2 , g3 , g4 , g5 } be the
reference set of students.
Let Ã be the fuzzy set of “smart” students,
where “smart” is fuzzy term.
Ã = {(g1 ,0.4)(g2 ,0.5)(g3 ,1)(g4 ,0.9)(g5
,0.8)}
Here Ã indicates that the smartness of g1 is
0.4 and so on.
Operations performed on fuzzy set:
1. Union
2. Intersection
3. Complement

Example of a Fuzzy Logic System
Let us consider an air conditioning system with 5-level fuzzy logic
system. This system adjusts the temperature of air conditioner by
comparing the room temperature and the target temperature value.
Algorithm:
1. Define linguistic Variables and terms
(start)
2. Construct membership functions for
them. (start)
3. Construct knowledge base of rules
(start)
4. Convert crisp data into fuzzy data sets
using membership functions.
(fuzzification)
5. Evaluate rules in the rule base.
from each rule.
(Inference Engine)
6. Combine results
(Inference Engine)
7. Convert output data into non-fuzzy
values. (defuzzification)

Example of a Fuzzy Logic System - Development
Step 1 − Define linguistic variables and terms
Linguistic variables are input and output variables in the form of simple words or sentences.
For room temperature, cold, warm, hot, etc., are linguistic terms.
Temperature (t) = {very-cold, cold, warm, very-warm, hot}
Every member of this set is a linguistic term and it can cover some portion of overall temperature values.
Step 2 − Construct membership functions for them
The membership functions of temperature variable are as shown ;

Step3 − Construct knowledge base rules
Create a matrix of room temperature values versus
target temperature values that an air conditioning
system is expected to provide.
Build a set of rules into the knowledge base in the
form of IF-THEN-ELSE structures.

Step 4 − Obtain fuzzy value
Fuzzy set operations perform evaluation of rules. The operations used for OR and AND are Max and Min
respectively. Combine all results of evaluation to form a final result. This result is a fuzzy value.
Step 5 − Perform defuzzification
Defuzzification is then performed according to membership function for output variable.

Application Areas of Fuzzy Logic
The key application areas of fuzzy logic are as
given;
1. Automotive Systems
 Automatic Gearboxes
 Four-Wheel Steering
 Vehicle environment control
2. Consumer Electronic Goods
 Hi-Fi Systems
 Photocopiers
 Still and Video Cameras
 Television
3. Domestic Goods
 Microwave Ovens
 Refrigerators
 Toasters
 Vacuum Cleaners
 Washing Machines
4. Environment Control
 Air Conditioners/Dryers/Heaters
 Humidifiers

Advantages and Disadvantages
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to complex
life, including
human
 Advantages of FLSs:
1. Mathematical concepts within fuzzy
reasoning are very simple.
2. You can modify a FLS by just adding or
deleting rules due to flexibility of fuzzy
logic.
3. Fuzzy logic Systems can take imprecise,
distorted, noisy input information.
4. FLSs are easy to construct and understand.
5. Fuzzy logic is a solution
problems in all fields of
medicine, as it resembles
reasoning and decision making.
 Disadvantages of FLSs:
1. There is no systematic approach to
fuzzy system designing.
2. They are understandable only when
simple.
3. They are suitable for the problems
which do not need high accuracy.

Statistical/ Probability Reasoning
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5
 Probabilistic reasoning is a way of knowledge representation where we apply the concept of
probability to indicate the uncertainty in knowledge.
 In probabilistic reasoning, we combine probability theory with logic to handle the uncertainty.
In the real world, there are lots of scenarios, where the certainty of something is not confirmed,
such as "It will rain today," "behavior of someone for some situations," "A match between two teams
or two players." These are probable sentences for which we can assume that it will happen but not
sure about it, so here we use probabilistic reasoning.
 Need of probabilistic reasoning in AI:
 When there are unpredictable outcomes.
 When specifications or possibilities of predicates becomes too large to handle.
 When an unknown error occurs during an experiment.
 In probabilistic reasoning, there are two ways to solve problems with uncertain knowledge:
1. Bayes' rule
2. Bayesian Statistics

Basic Terminologies
1. Probability: Probability can be defined as a chance that an uncertain event will occur. It is the
numerical measure of the likelihood that an event will occur.
 The value of probability always remains between 0 and 1 that represent ideal uncertainties.
1. 0 ≤ P(A) ≤ 1, where P(A) is the probability of an event A.
2. P(A) = 0, indicates total uncertainty in an event A.
3. P(A) =1, indicates total certainty in an event A.
 We can find the probability of an uncertain event by using the below formula.
2. Event: Each possible outcome of a variable is called an event.
3. Sample space: The collection of all possible events is called sample space.

Basic Terminologies
4. Random variables: Random variables are used to represent the events and objects in the real
world.
5. Prior probability: The prior probability of an event is probability computed before observing new
information.
6. Posterior Probability: The probability that is calculated after all evidence or information has taken
into account. It is a combination of prior probability and new information.
7.Conditional probability: Conditional probability is a probability of occurring an event when another
event has already happened. Let's suppose, we want to calculate the event A when event B has
already occurred, "the probability of A under the conditions of B", and If the probability of A is given
and we need to find the probability of B, it can be written as:
Where P(A⋀B)= Joint probability of a and B
P(B)= Marginal probability of B.

Conditional Probability Example
It can be explained by using the below Venn diagram, where B is
occurred event, so sample space will be reduced to set B, and
now we can only calculate event A when event B is already
occurred by dividing the probability of P(A⋀B) by P( B ).
Example:
In a class, there are 70% of the students who like English and 40% of the
students who likes English and mathematics, and then what is the
percent of students those who like English also like mathematics?
Solution:
Let, A is an event that a student likes Mathematics
B is an event that a student likes English.
Hence, 57% are the students who like English also like Mathematics.

Bayes' theorem in Artificial intelligence
 Bayes' theorem is also known as Bayes' rule, Bayes' law, or Bayesian reasoning, which determines
the probability of an event with uncertain knowledge.
 In probability theory, it relates the conditional probability and marginal probabilities of two random
events.
 Bayes' theorem was named after the British mathematician Thomas Bayes. The Bayesian
inference is an application of Bayes' theorem, which is fundamental to Bayesian statistics.
 It is a way to calculate the value of P(B|A) with the knowledge of P(A|B).

Bayes' theorem in Artificial intelligence
 The above equation (a) is called as Bayes' rule or Bayes' theorem. This equation is basic of most
modern AI systems for probabilistic inference. Where;
1. P(A|B) is known as posterior, which we need to calculate, and it will be read as Probability of
hypothesis A when we have occurred an evidence B.
2. P(B|A) is called the likelihood, in which we consider that hypothesis is true, then we calculate the
probability of evidence.
3. P(A) is called the prior probability, probability of hypothesis before considering the evidence
4. P(B) is called marginal probability, pure probability of an evidence.

Bayes' theorem Example

Applications
• Credit card fraud detection
• Spam filtering
• Medical diagnosis
• Patterns in customer dataset/marketing campaign performance
• Help robots make decisions
• Reconstructing clean images from noisy images
• Weather prediction
• Speech emotion recognition
• Optical character recognition
• Forensic analysis

Bayesian Belief Network in AI
Bayesian belief network is key computer technology for
dealing with probabilistic events and to solve a problem
which has uncertainty. We can define a Bayesian network as;
"A Bayesian network is a probabilistic graphical model
which represents a set of variables and their conditional
dependencies using a directed acyclic graph.“
 It is also called a Bayes network, belief network, decision
network, or Bayesian model.
Bayesian networks are probabilistic, because these
networks are built from a probability distribution, and also
use probability theory for prediction and anomaly detection.
It can also be used in various tasks including prediction,
anomaly detection, diagnostics, automated insight,
reasoning, time series prediction, and decision making
under uncertainty.
Bayesian Network can be used for building
models from data and experts opinions, and
it consists of two parts;
1. Directed Acyclic Graph
2. Table of conditional probabilities.
The generalized form of Bayesian network
that represents and solve decision problems
under uncertain knowledge is known as
an Influence diagram.
 A Bayesian network graph is made up of
nodes and Arcs (directed links), where;

Basic Terminologies of Bayesian Belief Network
 The Bayesian network has mainly two components:
1. Causal Component
2. Actual numbers
Each node in the Bayesian network has condition probability distribution P(Xi |Parent(Xi) ), which determines
the effect of the parent on that node.
 Bayesian network is based on Joint probability distribution and conditional probability.
 Joint probability distribution:
If we have variables x1, x2, x3,....., xn, then the probabilities of a different combination of x1, x2, x3.. xn, are
known as Joint probability distribution.
P[x1, x2, x3,....., xn], it can be written as the following way in terms of the joint probability distribution.
= P[x1| x2, x3,....., xn]P[x2, x3,....., xn]
= P[x1| x2, x3,....., xn]P[x2|x3,....., xn]....P[xn-1|xn] P[xn].
In general for each variable Xi, we can write the equation as:
P(Xi|Xi-1,........., X1) = P(Xi |Parents(Xi ))

Example
Harry installed a new burglar alarm at his home to detect
burglary. The alarm reliably responds at detecting a
burglary but also responds for minor earthquakes. Harry
has two neighbors David and Sophia, who have taken a
responsibility to inform Harry at work when they hear the
alarm. David always calls Harry when he hears the alarm,
but sometimes he got confused with the phone ringing
and calls at that time too. On the other hand, Sophia likes
to listen to high music, so sometimes she misses to hear
the alarm. Here we would like to compute the probability
of Burglary Alarm.
Problem:
Calculate the probability that alarm has sounded, but
there is neither a burglary, nor an earthquake occurred,
and David and Sophia both called the Harry.
Solution:
2K
1. The Bayesian network for the above problem is given
below. The network structure is showing that burglary
and earthquake is the parent node of the alarm and
directly affecting the probability of alarm's going off, but
David and Sophia's calls depend on alarm probability.
2. The network is representing that our assumptions do
not directly perceive the burglary and also do not notice
the minor earthquake, and they also not confer before
calling.
3. The conditional distributions for each node are given as
conditional probabilities table or CPT.
4. Each row in the CPT must be sum to 1 because all the
entries in the table represent an exhaustive set of cases
for the variable.
5. In CPT, a boolean variable with k boolean parents
contains probabilities. Hence, if there are two
parents, then CPT will contain 4 probability values.

Example
List of all events occurring in this network:
 Burglary (B)
 Earthquake(E)
 Alarm(A)
 David Calls(D)
 Sophia calls(S)
We can write the events of problem statement in the form of
probability: P[D, S, A, B, E], can rewrite the above probability
statement using joint probability distribution:
P[D, S, A, B, E] = P[D | S, A, B, E]. P[S, A, B, E]
= P[D | S, A, B, E]. P[S | A, B, E]. P[A, B, E]
= P [D| A]. P [ S| A, B, E]. P[ A, B, E]
= P[D | A]. P[ S | A]. P[A| B, E]. P[B, E]
= P[D | A ]. P[S | A]. P[A| B, E]. P[B |E]. P[E]

Example
Let's take the observed probability for the Burglary and earthquake component:
1. P(B= True) = 0.002, which is the probability of burglary.
2. P(B= False)= 0.998, which is the probability of no burglary.
3. P(E= True)= 0.001, which is the probability of a minor earthquake
4. P(E= False)= 0.999, Which is the probability that an earthquake not occurred.

Example
58
From the formula of joint distribution, we can write the problem statement in the form of probability
distribution:
P(S, D, A, ¬B, ¬E) = P (S|A) *P (D|A)*P (A|¬B ^ ¬E) *P (¬B) *P (¬E).
= 0.75* 0.91* 0.001* 0.998*0.999
= 0.00068045.
Hence, a Bayesian network can answer any query about the domain by using Joint distribution.
The semantics of Bayesian Network:
There are two ways to understand the semantics of the Bayesian network, which is given below:
1. To understand the network as the representation of the Joint probability distribution.
It is helpful to understand how to construct the network.
2. To understand the network as an encoding of a collection of conditional independence statements.
It is helpful in designing inference procedure.

References 50
• “Artificial intelligence: A modern approach”, S. J. Russell and P. Norvig,
Pearson, 3rd Edition
• “Artificial Intelligence”, Elaine R. and Kevin K., McGraw Hill, 2nd edition

Thank you
Sanjivani College of Engineering, Kopargaon Dept of Information Technology K. S. Ubale

Unit 6 Uncertainty.pptx

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