Transcript Lecture 13 - Suraj @ LUMS

Planning Chapter 7
article 7.4
Production Systems Chapter 5 article 5.3
RBS Chapter 7
article 7.2
RBS
RBS: Handling Uncertainties
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How to handle vague concepts?
Why vagueness occurs?
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All rules are not 100% deterministic
Certain rules are often true but not always
Headache may be caused in flu, but may not
always occur
An expert may not always be sure about
certain relations and associations
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RBS
First Source of Uncertainty:
The Representation Language
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
There are many more states of the real world than
can be expressed in the representation language
So, any state represented in the language may
correspond to many different states of the real
world, which the agent can’t represent distinguishably
On(A,B)  On(B,Table)  On(C,Table)  Clear(A)  Clear(C)
A
B
A
C
C
B
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A
B
C
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First Source of Uncertainty:
RBS
The Representation Language
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6 propositions On(x,y), where x, y = A, B, C and x  y
3 propositions On(x,Table), where x = A, B, C
3 propositions Clear(x), where x = A, B, C
At most 212 states can be distinguished in the
language [in fact much fewer, because of state
constraints such as On(x,y)  On(y,x)]
But there are infinitely many states of the real world
On(A,B)  On(B,Table)  On(C,Table)  Clear(A)  Clear(C)
A
B
A
C
C
B
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B
C
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RBS
Second source of Uncertainty:
Imperfect Observation of the World
Observation of the world can be:
 Partial, e.g., a vision sensor can’t see
through obstacles (lack of percepts)
R1
R2
The robot may not know whether
there is dust in room R2
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Second source of Uncertainty:
Imperfect Observation of the World
Observation of the world can be:
 Partial, e.g., a vision sensor can’t see
through obstacles
 Ambiguous, e.g., percepts have multiple
possible interpretations
A
C
B
On(A,B)  On(A,C)
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Second source of Uncertainty:
Imperfect Observation of the World
Observation of the world can be:
 Partial, e.g., a vision sensor can’t see
through obstacles
 Ambiguous, e.g., percepts have multiple
possible interpretations
 Incorrect
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RBS
Third Source of Uncertainty:
Ignorance, Laziness, Efficiency
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An action may have a long list of preconditions, e.g.:
Drive-Car:
P = Have(Keys)  Empty(Gas-Tank)  Battery-Ok 
Ignition-Ok  Flat-Tires  Stolen(Car) ...
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The agent’s designer may ignore some preconditions
... or by laziness or for efficiency, may not want to
include all of them in the action representation
The result is a representation that is either
incorrect – executing the action may not have the
described effects – or that describes several
alternative effects
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Representation of Uncertainty
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Many models of uncertainty
We will consider two important models:
•
•
Non-deterministic model:
Uncertainty is represented by a set of possible
values, e.g., a set of possible worlds, a set of
possible effects, ...
Probabilistic model:
Uncertainty is represented by a probabilistic
distribution over a set of possible values
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RBS
Example: Belief State
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In the presence of non-deterministic sensory
uncertainty, an agent belief state represents all the
states of the world that it thinks are possible at a
given time or at a given stage of reasoning
In the probabilistic model of uncertainty, a probability
is associated with each state to measure its likelihood
to be the actual state
0.2
0.3
0.4
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0.1
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RBS
What do probabilities mean?
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Probabilities have a natural frequency interpretation
The agent believes that if it was able to return many
times to a situation where it has the same belief state,
then the actual states in this situation would occur at
a relative frequency defined by the probabilistic
distribution
0.2
0.3
0.4
0.1
This state would occur
20% of the times
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RBS
Example
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Consider a world where a dentist agent D meets a new
patient P
D is interested in only one thing: whether P has a cavity,
which D models using the proposition Cavity
Before making any observation, D’s belief state is:
Cavity
p

 Cavity
1-p
This means that if D believes that a fraction p of
patients have cavities
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Where do probabilities come
from?
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Frequencies observed in the past, e.g., by the agent, its
designer, or others
Symmetries, e.g.:
•
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If I roll a dice, each of the 6 outcomes has probability 1/6
Subjectivism, e.g.:
•
•
If I drive on Highway 280 at 120mph, I will get a speeding
ticket with probability 0.6
Principle of indifference: If there is no knowledge to consider
one possibility more probable than another, give them the same
probability
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RBS
Expert System:
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A SYSTEM that mimics a human expert
Human experts always have in most case
some vague (not very precise) ideas
about the associations
Handling uncertainties is a essential part
of an expert system
Expert systems are RBS with some level
of uncertainty incorporated in the system
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RBS
Choosing a Problem
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Costs:
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Technical Problems:
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Choose problems that justify the development cost of
the expert systems
Choose a problem that is highly technical in nature
problems with Well-formed knowledge are the best
choice.
Highly specialized expert requirements, solution time for
the problem is not short time.
Cooperation from an expert:
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Experts are willingly to participate in the activity.
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Choosing a Problem
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Problems that are not suitable
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Problems for which experts are not available at
all, or they are not willingly to participate
Problems in which high common sense
knowledge is involved
Problems which involve high physical skills
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RBS
ES Architecture
Explanation
system
interface
user
Inference
engine
Knowledge
Base
editor
Case
specific
Data
Knowledge
Base
Expert System Shell
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ES Architecture
Uses Menus, NLP, etc…
Which is used to interact
With the users
Explanation
system
interface
user
Inference
engine
Knowledge
Base
editor
Case
specific
Data
Knowledge
Base
Expert System Shell
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ES Architecture
Implements the
reasoning methods
Generally backward chaining
Explains why a decision
is taken, uses keywords
Such as HOW, WHY etc…
Explanation
system
interface
user
Updates the KB
Inference
engine
Knowledge
Base
editor
Case
specific
Data
Knowledge
Base
Expert System Shell
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ES Architecture
Pre-solved problems,
Frequently referred cases
Explanation
system
interface
user
Inference
engine
Knowledge
Base
editor
Expert System Shell
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Case
specific
Data
Knowledge
Base
Collection of facts
And rules
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RBS
Shells
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General purpose toolkit/shell is problem
independent
Shells commercially available
CLIPS is one such shell
Freely available
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Reasoning with Uncertainty
Case Studies:
 MYCIN
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Implements certainty factors approach
INTERNIST: Modeling Human Problem
Solving
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Implements Probability approach
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Probability based ES
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Probability:
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Degree of believe in a fact ‘x’, P(x)
P(H): degree of believe in H, when
supporting evidence is NOT given, H is
the hypothesis
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P(H|E): degree of believe in H, when
supporting evidence is given, E is the
evidence supporting hypothesis
P(H|E): conditional probability
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Conditional Probability
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P(H|E): conditional probability is given
through a LAW (RULE)
P(H|E)=P(H^E)/P(E)
where P(H^E) is the probability of H and
E occurring together (both are TRUE)
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RBS
Evaluating: Conditional Probability
P(H|E): P(Heart Attack|shooting arm pain)
Two approaches can be adopted:
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Asking an expert about the frequency of it
happening
Finding the probability from the given data
Second Approach
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Collect the data for all the patients
complaining about the shooting arm pain.
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Find the proportion of the patients that had
an heart attack from the data collected in
step 1
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RBS
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Evaluating: Conditional Probability
P(H|E): P(Heart Attack|shooting arm pain)
Probability of Disease given symptoms
VS
P(E|H): P(shooting arm pain|Heart Attack)
Probability of symptoms given Disease
Which is easier to find of the two?
Perhaps P(E|H) is easier
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RBS
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Evaluating: Conditional Probability
P(H|E): P(Heart Attack|shooting arm pain)
Probability of Disease given symptoms
Headache is mostly experienced when a
patient suffers from flu, fever, tumor, etc…
Find out whether a patient suffers from
tumor, given headache
Collect the data for all the headache
patients, and then find the proportion of
patients that have tumor.
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RBS
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Evaluating: Conditional Probability
P(E|H): P(shooting arm pain|Heart Attack)
Probability of symptoms given Disease
Headache is mostly experienced when a
patient suffers from flu, fever, tumor, etc…
Find out whether a tumor patient suffers
from headache
Collect the data for all the tumor patients,
and then find the proportion of patients
that have headache
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RBS
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Evaluating: Conditional Probability
Generally speaking P(E|H):
P(shooting arm pain|Heart Attack) is
easier to find.
Therefore the we need to convert
P(H|E) in terms of P(E|H)
P(H|E)=P(H^E)/P(E)
P(H|E)=[P(E|H)*P(H)]/P(E)
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RBS
Evaluating: Conditional Probability
More than one evidence
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Independence of events
P(H|E1^E2)=P(H^E1^E2)/P(E1^E2)
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P(H|E1^E2)=[P(E1|H)* P(E2|H)* P(H)]/P(E1)*P(E2)
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RBS
Inference through Joint Prob.
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Start with the joint probability distribution:
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Inference by enumeration
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Start with the joint probability distribution:
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P(toothache) = 0.108 + 0.012 + 0.016 + 0.064 =
0.2
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RBS
Inference by enumeration
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Start with the joint probability distribution:
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P(toothache) = 0.108 + 0.012 + 0.016 + 0.064 = 0.2
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RBS
Inference by enumeration
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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
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RBS
Certainty Factors (CF)
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CF for rules CF(R)
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CF for Pre-conditions CF(PC)
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From the experts
From the end user
CF for conclusions CF(cl)
CF(cl)=CF(R)*CF(PC)
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Certainty Factors (CF)
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CF for rules CF(R)
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CF(R) = 0.6
CF for Pre-conditions CF(PC)
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IF A then B
IF A (0.4) then B CF(A)= 0.4
CF for conclusions CF(cl)
CF(B)=CF(R)*CF(A)= 0.6*0.4=0.24
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RBS
Finding Overall CF for PC
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If A(0.1) and B(0.4) and C(0.5) Then D
Overall CF(PC)=min(CF(A),CF(B),CF(C))
CF(PC)=0.1
If A(0.1) or B(0.4) or C(0.5) Then D
Overall CF(PC)=max(CF(A),CF(B),CF(C))
CF(PC)=0.5
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RBS
Combining Certainty factors
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When the conclusions are same and certainty
factors are positive:
CF(R1)+CF(R2) – CF(R1)*CF(R2)
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When the conclusions are same and the certainty
factors are both negative
CF(R1)+CF(R2) + CF(R1)*CF(R2)
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Otherwise: both conclusions are same but have
different signs
[CF(R1)+CF(R2)] / [1 – min ( | CF(R1) | , | CF(R1) |]
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Example
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Please see the class handouts
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