Download Search problems

Uncertainty
Russell and Norvig: Chapter 14, 15
Koller article on BNs
CMCS424 Spring 2002
April 23
Uncertain Agent
sensors
?
?
environment
agent
?
actuators
model
An Old Problem …
Types of Uncertainty
Uncertainty in prior knowledge
E.g., some causes of a disease are unknown and are
not represented in the background knowledge of a
medical-assistant agent
Types of Uncertainty
For example, to drive my car in the morning:
Uncertainty
in prior
knowledge
• It
must not have
been
stolen during the night
E.g., some causes of a disease are unknown and are
• It
must
not have in
flat
tires
not
represented
the
background knowledge of a
• There
must be gasagent
in the tank
medical-assistant
• The
battery must
not be dead
Uncertainty
in actions
E.g.,ignition
actions must
are represented
with relatively short lists
• The
work
preconditions,
while
lists are in fact arbitrary
• Iofmust
not have lost
thethese
car keys
long
• No truck should obstruct the driveway
• I must not have suddenly become blind or paralytic
Etc…
Not only would it not be possible to list all of them, but
would trying to do so be efficient?
Types of Uncertainty
Uncertainty in prior knowledge
E.g., some causes of a disease are unknown and are
not represented in the background knowledge of a
medical-assistant agent
Uncertainty in actions
E.g., actions are represented with relatively short lists
of preconditions, while these lists areCourtesy
in factR.arbitrary
Chatila
long
Uncertainty in perception
E.g., sensors do not return exact or complete
information about the world; a robot never knows
exactly its position
Types of Uncertainty
Uncertainty in prior knowledge
E.g., some causes of a disease are unknown and are
Sources
ofbackground
uncertainty:
not represented
in the
knowledge of a
medical-assistant
agent
1. Ignorance
Uncertainty in actions
E.g., actions
are represented
with relatively short lists
2. Laziness
(efficiency?)
of preconditions, while these lists are in fact arbitrary
long
Uncertainty in perception
E.g.,
sensors
do not return exactisoracomplete
What
we
call uncertainty
summary
information about the world; a robot never knows
of all
thatitsisposition
not explicitly taken into account
exactly
in the agent’s KB
Questions
How to represent uncertainty in
knowledge?
How to perform inferences with
uncertain knowledge?
Which action to choose under
uncertainty?
How do we deal with
uncertainty?
Implicit:


Ignore what you are uncertain of when you can
Build procedures that are robust to uncertainty
Explicit:


Build a model of the world that describe
uncertainty about its state, dynamics, and
observations
Reason about the effect of actions given the
model
Handling Uncertainty
Approaches:
1. Default reasoning
2. Worst-case reasoning
3. Probabilistic reasoning
Default Reasoning
Creed: The world is fairly normal.
Abnormalities are rare
So, an agent assumes normality, until
there is evidence of the contrary
E.g., if an agent sees a bird x, it assumes
that x can fly, unless it has evidence that
x is a penguin, an ostrich, a dead bird, a
bird with broken wings, …
Representation in Logic
BIRD(x)  ABF(x)  FLIES(x)
Very active research
field
in the 80’s
PENGUINS(x)
 AB
(x)
F
 Non-monotonic logics: defaults, circumscription,
BROKEN-WINGS(x)

AB
(x)
F
closed-world assumptions
BIRD(Tweety)
Applications to databases
…
Default rule: Unless ABF(Tweety) can be proven
True, assume it is False
But what to do if several defaults are contradictory?
Which ones to keep? Which one to reject?
Worst-Case Reasoning
Creed: Just the opposite! The world is ruled
by Murphy’s Law
Uncertainty is defined by sets, e.g., the set
possible outcomes of an action, the set of
possible positions of a robot
The agent assumes the worst case, and
chooses the actions that maximizes a utility
function in this case
Example: Adversarial search
Probabilistic Reasoning
Creed: The world is not divided between
“normal” and “abnormal”, nor is it
adversarial. Possible situations have
various likelihoods (probabilities)
The agent has probabilistic beliefs –
pieces of knowledge with associated
probabilities (strengths) – and chooses
its actions to maximize the expected
value of some utility function
How do we represent Uncertainty?
We need to answer several questions:
What do we represent & how we represent it?

What language do we use to represent our
uncertainty? What are the semantics of our
representation?
What can we do with the representations?

What queries can be answered? How do we
answer them?
How do we construct a representation?

Can we ask an expert? Can we learn from data?
Probability
A well-known and well-understood framework
for uncertainty
Clear semantics
Provides principled answers for:



Combining evidence
Predictive & Diagnostic reasoning
Incorporation of new evidence
Intuitive (at some level) to human experts
Can be learned
Notion of Probability
P(AvA)
= P(A)+P(A)-P(A
A)
You drive on Rt 1 to UMD
often, and
you notice that 
70%
P(True)
= P(A)+P(A)-P(False)
of the times there is a traffic
slowdown
at the intersection
= P(A)
P(A)
of PaintBranch & Rt 1. The next1 time
you+plan
to drive on
So:the proposition “there is a
Rt 1, you will believe that
P(A) =of1PB
- P(A)
slowdown at the intersection
& Rt 1” is True with
probability 0.7
The probability of a proposition A is a real
number P(A) between 0 and 1
P(True) = 1 and P(False) = 0
P(AvB) = P(A) + P(B) - P(AB)
Axioms of probability
Frequency Interpretation
Draw a ball from a bag containing n balls
of the same size, r red and s yellow.
The probability that the proposition A =
“the ball is red” is true corresponds to
the relative frequency with which we
expect to draw a red ball  P(A) = r/n
Subjective Interpretation
There are many situations in which there
is no objective frequency interpretation:


On a windy day, just before paragliding from
the top of El Capitan, you say “there is
probability 0.05 that I am going to die”
You have worked hard on your AI class and
you believe that the probability that you will
get an A is 0.9
Random Variables
A proposition that takes the value True with
probability p and False with probability 1-p is
a random variable with distribution (p,1-p)
If a bag contains balls having 3 possible
colors – red, yellow, and blue – the color of a
ball picked at random from the bag is a
random variable with 3 possible values
The (probability) distribution of a random
variable X with n values x1, x2, …, xn is:
(p1, p2, …, pn)
with P(X=xi) = pi and Si=1,…,n pi = 1
Joint Distribution
k random variables X1, …, Xk
The joint distribution of these variables is a
table in which each entry gives the probability
of one combination of values of X1, …, Xk
Example:
Toothache
Toothache
0.04
0.06
Cavity 0.01
0.89
Cavity
P(CavityToothache)
P(CavityToothache)
Joint Distribution Says It All
Toothache
Toothache
0.04
0.06
Cavity 0.01
0.89
Cavity
P(Toothache) = P((Toothache Cavity) v (ToothacheCavity))
= P(Toothache Cavity) + P(ToothacheCavity)
= 0.04 + 0.01 = 0.05
P(Toothache v Cavity)
= P((Toothache Cavity) v (ToothacheCavity)
v (Toothache Cavity))
= 0.04 + 0.01 + 0.06 = 0.11
Conditional Probability
Definition:
P(A|B) =P(AB) / P(B)
Read P(A|B): probability of A given B
can also write this as:
P(AB) = P(A|B) P(B)
called the product rule
Example
Toothache
Toothache
0.04
0.06
Cavity 0.01
0.89
Cavity
P(Cavity|Toothache) = P(CavityToothache) / P(Toothache)
 P(CavityToothache) = ?
 P(Toothache) = ?
 P(Cavity|Toothache) = 0.04/0.05 = 0.8
Generalization
P(A  B  C) = P(A|B,C) P(B|C) P(C)
Bayes’ Rule
P(A  B) = P(A|B) P(B)
= P(B|A) P(A)
P(A|B) P(B)
P(B|A) =
P(A)
Example
Toothache
Toothache
0.04
0.06
Cavity 0.01
0.89
Cavity
Representing Probability
Naïve representations of probability run into
problems.
Example:
 Patients in hospital are described by several
attributes:
 Background: age, gender, history of diseases, …
 Symptoms: fever, blood pressure, headache, …
 Diseases: pneumonia, heart attack, …
A probability distribution needs to assign a number to
each combination of values of these attributes


20 attributes require 106 numbers
Real examples usually involve hundreds of attributes
Practical Representation
Key idea -- exploit regularities
Here we focus on exploiting
conditional independence properties
Independent Random
Variables
Two variables X and Y are independent if


P(X = x|Y = y) = P(X = x) for all values x,y
That is, learning the values of Y does not change
prediction of X
If X and Y are independent then

P(X,Y) = P(X|Y)P(Y) = P(X)P(Y)
In general, if X1,…,Xn are independent, then


P(X1,…,Xn)= P(X1)...P(Xn)
Requires O(n) parameters
Conditional Independence
Unfortunately, random variables of interest
are not independent of each other
A more suitable notion is that of conditional
independence
Two variables X and Y are conditionally
independent given Z if



P(X = x|Y = y,Z=z) = P(X = x|Z=z) for all values x,y,z
That is, learning the values of Y does not change prediction
of X once we know the value of Z
notation: Ind( X ; Y | Z )
Car Example
Three propositions:



Gas
Battery
Starts
P(Battery|Gas) = P(Battery)
Gas and Battery are independent
P(Battery|Gas,Starts) ≠ P(Battery|Starts)
Gas and Battery are not independent given
Starts
Example: Naïve Bayes Model
A common model in early diagnosis:

Symptoms are conditionally independent given the
disease (or fault)
Thus, if


X1,…,Xn denote whether the symptoms exhibited
by the patient (headache, high-fever, etc.) and
H denotes the hypothesis about the patients
health
then, P(X1,…,Xn,H) = P(H)P(X1|H)…P(Xn|H),
This naïve Bayesian model allows compact
representation

It does embody strong independence assumptions
Markov Assumption
We now make this
independence assumption
more precise for
directed acyclic
graphs (DAGs)
Each random variable X,
is independent of its nondescendents, given its
parents Pa(X)
Formally,
Ind(X; NonDesc(X) |
Pa(X))
Ancestor
Parent
Y1
Y2
X
Non-descendent
Descendent
Markov Assumption Example
Earthquake
In this example:





Ind(
Ind(
Ind(
Ind(
Ind(
E; B )
B; E, R )
R; A, B, C | E )
A; R | B,E )
C; B, E, R | A)
Radio
Burglary
Alarm
Call
I-Maps
A DAG G is an I-Map of a distribution P
if the all Markov assumptions implied by
G are satisfied by P
Examples:
X
Y
X
Y
Factorization
Given that G is an I-Map of P, can we simplify
the representation of P?
Example:
X
Y
Since Ind(X;Y), we have that P(X|Y) = P(X)
Applying the chain rule
P(X,Y) = P(X|Y) P(Y) = P(X) P(Y)
Thus, we have a simpler representation of
P(X,Y)
Factorization Theorem
Thm: if G is an I-Map of P, then
P( X1 ,..., X n )   P( X i | Pa( X i ))
i
Factorization Example
Earthquake
Radio
Burglary
Alarm
Call
P(C,A,R,E,B) =
P(B)P(E|B)P(R|E,B)P(A|R,B,E)P(C|A,R,B,E)
versus
P(C,A,R,E,B) = P(B) P(E) P(R|E) P(A|B,E)
P(C|A)
Consequences
We can write P in terms of “local” conditional
probabilities
If G is sparse,

that is, |Pa(Xi)| < k ,
 each conditional probability can be specified
compactly

e.g. for binary variables, these require O(2k)
params.
 representation of P is compact

linear in number of variables
Bayesian Networks
A Bayesian network specifies a probability
distribution via two components:


A DAG G
A collection of conditional probability distributions
P(Xi|Pai)
The joint distribution P is defined by the
factorization
P (X1 ,..., Xn )   P (Xi | Pai )
i
Additional requirement: G is a minimal I-Map
of P
Bayesian Networks
Pneumonia
nodes = random variables
edges = direct probabilistic
influence
Tuberculosis
Lung Infiltrates
XRay
Sputum Smear
Network structure encodes independence assumptions:
XRay conditionally independent of Pneumonia given Infiltrates
Bayesian Networks
P T P(I |P, T )
p t 0.8 0.2
p t
0.6 0.4
p t
0.2 0.8
Pneumonia
Tuberculosis
Lung Infiltrates
p t 0.01 0.99
XRay
Sputum Smear
Each node Xi has a conditional probability
distribution P(Xi|Pai)


If variables are discrete, P is usually multinomial
P can be linear Gaussian, mixture of Gaussians, …
BN Semantics
P
T
conditional
local
full joint
independencies + probability = distribution
I
models
over domain
in
BN
structure
X
S
P(p ,t,i, x, s )  P(p ) P(t) P(i | p ,t)
P(x | i) P(s | t)
Compact & natural representation:
k
n params
 nodes have  k parents  2 n vs. 2
Queries
Full joint distribution specifies answer to any query:
P(variable | evidence about others)
Pneumonia
Tuberculosis
Lung Infiltrates
XRay
Sputum Smear
BN Learning
P
Inducer
Data
T
I
X
S
BN models can be learned from empirical data


parameter estimation via numerical optimization
structure learning via combinatorial search.
BN hypothesis space biased towards distributions
with independence structure.
Questions
How to represent uncertainty in
knowledge?
If a goal is terribly important, an agent may be better
to aperform
inferences
with action
offHow
choosing
less efficient,
but less uncertain
than
a more efficient
one
uncertain
knowledge?
Which action to choose under
uncertainty?
But if the goal is also extremely urgent, and the less
uncertain action is deemed too slow, then the agent may
take its chance with the faster, but more uncertain action
Summary
Types of uncertainty
Default/worst-case/probabilistic reasoning
Probability Theory
Bayesian Networks
Making decisions under uncertainty
Exciting Research Area!
References
Russell & Norvig, chapters 14, 15
Daphne Koller’s BN notes, available from the
class web page
Jean-Claude Latombe’s excellent lecture
notes,
http://robotics.stanford.edu/~latombe/cs121/w
inter02/home.htm
Nir Friedman’s excellent lecture notes,
http://www.cs.huji.ac.il/~pmai/
Questions
How to represent uncertainty in
knowledge?
How to perform inferences with
uncertain knowledge?
When a doctor receives lab analysis results for some
patient, how do they change his prior knowledge
about the health condition of this patient?
Example: Robot Navigation
Courtesy S. Thrun
Uncertainty in control
(Control uncertainty cone)
Worst-Case Planning
control
uncertainty cone
robot
initial position
uncertainty
power
station
Target Tracking Example
robot
target
1. Open-loop vs. closed-loop strategy
Target Tracking Example
robot
target
1. Open-loop vs. closed-loop strategy
2. Off-line vs. on-line planning/reasoning
Target Tracking Example
robot
target
Utility =
escape time
Of the target
1. Open-loop vs. closed-loop strategy
2. Off-line vs. on-line planning/reasoning
3. Maximization of worst-case value of utility
vs. of expected value of utility