Lesson04-Uncertainty - Pt. 1 Probabilistic Methods.pptx

Lesson 04: Uncertainty
Pt.1 Probabilistic Methods
Uncertainty, belief, probability, fuzziness

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IA / 4 Uncertainty 3
Uncertain (adjective)
 doubtful
 a: not known beyond doubt
 b: not having certain knowledge
 c: not clearly identified or defined
 not constant
 indefinite, indeterminate
 not certain to occur
 not reliable, untrustworthy

Content
 The problem of uncertainty
 Handling uncertainty
 Probabilistic Methods
 Fuzzy Methods

From clean worlds to noisy worlds
 The AI methods we have seen so far have
representations that perfectly match their “worlds”
 Inferences by sound reasoning can then be translated
to the real world and they make sense there
 Unfortunately, this is not
always the situation that
we have in real-life
 World is noisy, perception
is noisy, reasoning is noisy

An historical example of the problem
 Expert Systems leverage expert decision making in a
limited domain (medical diagnosis, computer
configuration, machine fault diagnosis, etc.).
 Knowledge is extracted from a willing expert
 Knowledge is usually represented as inference rules
 These systems can replace human experts in their
narrow, precise, domains of expertise

The MYCIN Expert System
 An AI program for diagnosis and therapy selection for
bacterial infections of the blood
 Provide advice to non-expert physicians with time
considerations and incomplete evidence
 Example MYCIN rule:
if stain of organism is gramneg and
morphology is rod and
aerobicity is aerobic
then strongly suggestive (0.8) that
organism is enterobacteriaceae.

MYCIN Architecture
Consultation
System
Explanation
System
Knowledge
Acquisition
System
Q-A System
Dynamic DB
Patient Data
Context Tree
Dynamic Data
Static DB
Rules
Parameter Properties
Context Type Properties
Tables, Lists
Physician
Expert

Knowledge and Reasoning process
 MYCIN domain-specific knowledge in over 450 rules
 Each rule is completely modular, all relevant context is
contained in the rule with explicitly stated premises
 Inexact reasoning uses certainty factors (CFs):
 Facts and rules have assigned confidences, from 0 to 1.
 Truth of a hypothesis is measured by a sum of the CFs
 Positive sum is confirming evidence
 Negative sum is disconfirming evidence
 MYCIN handled uncertainty. To an extent.

Different problems to consider
 Imprecision and accuracy
 Lack of exactness or accuracy
 “The result was 13.8 ± 0.3”
 Vagueness
 Lack of certainty or distinctness
 Lack of preciseness in thought or communication
 “The result was quite good”
 Uncertainty
 Not able to be relied on; not known or definite
 Not completely confident or sure of something
 “I believed that the result was 13”
Precision:
precisión
Accuracy:
exactitud

Uncertainty in AI
 Uncertainty is pervasive in “real-world” AI
 AI agents suffer uncertainty in:
 Perception
 Representation
 Reasoning
 Action
 In this lesson we will show a couple of methods to deal
with uncertainty: probabilistic and fuzzy
Thought
Representation

Limits of (classical) logics
 Trying to use logic to cope with a real-world domain like
medical diagnosis fails for three main reasons:
 Laziness/time/cost: It is too much work to list the complete set of
antecedents or consequents needed to ensure an exceptionless rule and
too hard to use such rules.
 Theoretical ignorance: Medical science –or automobile engineering– has
no complete, usable, theory for the domain.
 Practical ignorance: Even if we know all the rules, we might be uncertain
about a particular patient because not all the necessary tests have been
or can be run or have noise or …
 c.f. the frame, qualification and ramification problems

Elimination of simplifying limitations
 In previous lessons we have seen fully observable,
deterministic, static, known environments
 In these worlds the solution to a problem is a precise
sequence of actions.
 For AIs addressing the real world we shall relax those
constraints: real world is not so neat.
 Agents in the real world need to handle uncertainty,
whether due to partial observability, nondeterminism,
dynamism, or the existence of adversaries.

Two Classes of Uncertainty
 Uncertainty –messiness in the mind of the agent– can
have two origins:
 Ontological: The world is messy
 Epistemological: The world is neat but
the agent can’t get a perfect picture
of it*
Representation
Real-world
* Neat in terms of the representation formalism

Ontological
Epistemological
a dimly seen rabbit a Schrödinger cat

Copilot, please make a
poem about a dimly seen
rabbit using quantum
mechanics terms
In the twilight's gentle embrace,
A rabbit's form, a fleeting trace,
In quantum realms, it hops and plays,
In superposition, it spends its days.
With wavefunctions that softly sway,
Its path uncertain, it cannot stay,
Entangled with the moonlit night,
A particle of pure delight.
Through Heisenberg's uncertain veil,
Its position and momentum frail,
A quantum leap, a boundless flight,
In the dimly seen, it finds its light.
In Schrödinger's box, it might reside,
Both here and there, it does not hide,
A paradox in every hop,
In quantum fields, it cannot stop.
So let us ponder, let us dream,
Of rabbits in a quantum scheme,
Where physics and the night entwine,
In a dance both wondrous and divine.
“Through Heisenberg's uncertain veil”

We’ll see two strategies
 Probabilistic methods
 Deal with epistemological uncertainty
 Fuzzy methods
 Deal with ontological uncertainty
 Other:
 Necessity/Possibility theory
 Depmster/Shafer belief functions theory
 Plaussibility theory
 Qualitative / Interval logics
 …

Dealing with degrees of belief
 In general, when there is a connection between real-
world facts (cause  effect, if P then Q), there are no
strict logical consequences in other directions.
The world is complex (if P R
∧ S
∧ T
∧ …
∧ then Q).
 the qualification problem
 Rain  Wet road
 Wet road  Rain
I notice the road is
wet.
Could it have rained?

Knowledge and Belief
 Philosophers said that
knowledge is “justified, true, belief”; however, …
 The agent’s “knowledge” can at best provide only a
degree of belief in the relevant sentences.
 Our main tool for dealing with degrees of belief is
probability theory.
 Probability theory talks also about “possible worlds”.

Probability theory commitments
 Probability provides a way of summarizing the
uncertainty that comes from our laziness and
ignorance, thereby “solving” the qualification problem.
 Commitments in logic and probability:
 The ontological commitments are the same—that the world is composed
of facts that hold or not.
 The epistemological commitments are different: a logical agent believes
each sentence to be {true, false, no opinion} whereas a probabilistic
agent may have a numerical degree of belief between 0 (for sentences
that are certainly false) and 1 (certainly true).

Fundamental concepts in probability
 Basics of probability:
 Random Variables
 Joint and Marginal Distributions
 Conditional Distribution
 Product Rule, Chain Rule, Bayes’ Rule
 Inference
 Independence
 Uses: Bayesian networks, Markov chains, etc.
 This is basic probability stuff (statistics) that is used a
lot in current AI and robotics methods

Probabilitstic Inference in Ghostbusters
 A ghost is hidden in the grid somewhere
 Sensor readings tell how close a square is to the ghost
 On the ghost: red
 1 or 2 away: orange
 3 or 4 away: yellow
 5+ away: green
 If sensors are noisy, we may
knowP(Color | Distance)
P(red | 3) P(orange | 3) P(yellow | 3) P(green | 3)
0.05 0.15 0.5 0.3

Dealing with Uncertainty
 General situation:
 Observed variables (evidence): Agent knows certain things
about the state of the world (e.g., sensor readings or
symptoms)
 Unobserved variables: Agent needs to reason about other
aspects (e.g. where an object is or what disease is present)
 Model: Agent knows something about how the known
variables relate to the unknown variables
 Probabilistic reasoning gives us a framework
for managing our beliefs (e.g. with new info)

Random Variables
 A random variable is some aspect of the world about
which we (may) have uncertainty
 R = Is it raining?
 T = Is it hot or cold?
 D = How long will it take to drive to work?
 L = Where is the ghost?
 Random variables have ranges (maybe non-numeric)
 R in {true, false} (often writen as {+r, -r})
 T in {hot, cold}
 D in [0, )
 L in possible locations, maybe {(0,0), (0,1), …}
We denote random
variables with upper
case letters: R,T,…

Probability Distributions
Random variables have probability distributions
Associate a probability with each value in the range
T P
hot 0.5
cold 0.5
W P
sun 0.6
rain 0.1
fog 0.3
meteor 0.0
Temperature
Weather
A probability value for each “possible world”

Ranges may be multidimensional
 A two dimensional
normal distribution
Ranges may also be discrete or continuous

Shorthand notation:
Probability Distributions
 A distribution is a function from
values to probabilities.
 A probability (lower case) is a
single number
p =
 Axioms for a valid “probability
distribution”:

Joint Distributions
 A joint distribution over a set of random variables
specifies a real number for each tuple (or outcome):
 Must obey:
T W P
hot sun 0.4
hot rain 0.1
cold sun 0.2
cold rain 0.3
A probabilistic model
is a joint distribution

Events
 An event is a set E of outcomes
 From a joint distribution, we can
calculate the probability of any event
 Probability that it’s hot AND sunny? 0.4
 Probability that it’s hot? 0.4+0.1 = 0.5
 Probability that it’s hot OR sunny? 0.4+0.1+0.2=0.7
 Typically, the events we care about are
partial assignments, like P(W=sunny)
T W P
hot sun 0.4
hot rain 0.1
cold sun 0.2
cold rain 0.3
“W=sunny” event
P(W=sunny) = 0.6

Quiz: Events
 P(+x AND +y) ?
 P(+x) ?
 P(-y OR +x) ?
X Y P
+x +y 0.2
+x -y 0.3
-x +y 0.4
-x -y 0.1

Marginal Distributions
 Marginal distributions are sub-tables which eliminate variables
 Marginalization (summing out): Combine collapsed rows by
adding
T W P
hot sun 0.4
hot rain 0.1
cold sun 0.2
cold rain 0.3
T P
hot 0.5
cold 0.5
W P
sun 0.6
rain 0.4

Quiz: Compute Marginal Distributions
X Y P
+x +y 0.2
+x -y 0.3
-x +y 0.4
-x -y 0.1
X P
+x
-x
Y P
+y
-y

Quiz: Compute Marginal Distributions
X Y P
+x +y 0.2
+x -y 0.3
-x +y 0.4
-x -y 0.1
X P
+x 0.5
-x 0.5
Y P
+y 0.6
-y 0.4

Conditional Probabilities
 Conditional probability is the probability of a, given b
 What is the probability of “sun” if it is “cold”?
 What is the probability of a sunny day being cold (P(cold|sunny))?
 This is the definition of a conditional probability
 This is a simple relation between joint, conditional and
marginal probabilities
b is called “evidence”

Conditional Probabilities
 Use the simple relation between joint and conditional
probabilities (probability of a, given b) taken from the
definition of conditional probability
T W P
hot sun 0.4
hot rain 0.1
cold sun 0.2
cold rain 0.3
What is the probability of “sun” if it is “cold”?

Quiz: Conditional Probabilities
 P(+x | +y) ?
 P(-x | +y) ?
X Y P
+x +y 0.2
+x -y 0.3
-x +y 0.4
-x -y 0.1

Conditional Distributions
 Conditional distributions are probability distributions
over some variables given fixed values of others
T W P
hot sun 0.4
hot rain 0.1
cold sun 0.2
cold rain 0.3
W P
sun 0.8
rain 0.2
W P
sun 0.4
rain 0.6
Conditional
Distributions
Joint
Distribution

Normalization Trick (scale to 1.0)
T W P
hot sun 0.4
hot rain 0.1
cold sun 0.2
cold rain 0.3
W P
sun 0.4
rain 0.6

SELECT the joint
probabilities
matching the
evidence (T=cold)
Normalization Trick
T W P
hot sun 0.4
hot rain 0.1
cold sun 0.2
cold rain 0.3
W P
sun 0.4
rain 0.6
T W P
cold sun 0.2
cold rain 0.3
NORMALIZE the
selection
(make it sum 1.0)

Quiz: Normalization Trick
X Y P
+x +y 0.2
+x -y 0.3
-x +y 0.4
-x -y 0.1
SELECT the joint
probabilities
matching the
evidence
NORMALIZE the
selection
(make it sum 1.0)
 P(X | Y=-y) ?

Independence
 Two random variables are said to be independent if
they do not condition each other:
P(a|b)=P(a)
P(b|a)=P(b)
P(a∧b)=P(a)P(b)
 When it is available, independence can help in reducing
the size of the domain representation and the
complexity of the inference problem.

Probabilistic Inference
 Probabilistic inference is the computation of a specific
probability from other, known probabilities
(e.g. conditional from joint)
 We generally compute conditional probabilities
 P(on time | no reported accidents) = 0.90
 These represent the agent’s beliefs given the evidence
 Probabilities change with new evidence:
 P(on time | no accidents, 5 a.m.) = 0.95
 P(on time | no accidents, 5 a.m., raining) = 0.80
 Observing new evidence causes beliefs to be updated

Probabilistic Inference Rules
 From the definition of a conditional probability
 We can derive some inference rules:
 The product rule
 The chain rule
 Bayes’ rule

The Product Rule
 Compute joint distributions from conditional and
marginal distributions

Quiz: The Product Rule
R P
sun 0.8
rain 0.2
D W P
wet sun 0.1
dry sun 0.9
wet rain 0.7
dry rain 0.3
D W P
wet sun 0.08
dry sun 0.72
wet rain 0.14
dry rain 0.06
0.08
0.72
0.14
0,06
Sum = 1.0

The Chain Rule
 Compute joint distribution as an incremental product of
conditional distributions

The chain rule
P(6, 6, 6)?
P(J, J, J)?

Bayes’ Rule
 There are two ways to factor a joint distribution over
two variables:
 Dividing, we get Bayes rule:
 Why is this at all useful?
 Derive one conditional probability from its reverse (often one
conditional is tricky but the other one is simpler, e.g. due to causality).
 Foundation of many probabilistic inference systems

Inference with Bayes’ Rule
 Example: Computing diagnostic probability from causal
probability (cause  effect)
 Meningitis causes stiff neck (p=0.8)
 I have a stiff neck. Should I worry?
 What is the probability of having meningitis if I have a
stiff neck? P(+m|+s)

Inference with Bayes’ Rule
 Example: Diagnostic probability from causal probability:
 Example:
 M: meningitis, S: stiff neck
 Note: posterior probability of meningitis still very small
 Note: you should still get stiff necks checked out! Why?
Known data
= 0.007944
What is the probability of
having meningitis if I have a sti
neck? P(+m|+s)

Quiz: Bayes’ Rule
 Given:
 What is P(W | dry) ?
R P
sun 0.8
rain 0.2
D W P
wet sun 0.1
dry sun 0.9
wet rain 0.7
dry rain 0.3

Uses of probability
 Probabilistic reasoning:
 Bayesian Networks
 Probabilistic reasoning over time:
 Markov Models
 Decision making:
 Utility functions, single and multiobjective
Decision theory = probability theory + utility theory
 Markov Decision Processes
 Multiagent decision making:
 Game theory
Mathematics
Statistics
Economy

Bayesian networks
 Bayesian networks are a way to represent probabilistic
relationships to capture uncertain knowledge.
 They can be used to perform probabilistic inference:
 Exact probabilistic inference is computationally intractable in the worst
case, but can be used in many practical situations.
 If not, approximate inference algorithms are often applicable.
 Other names for Bayesian networks:
 They were called belief networks in the 1980s and 1990s.
 A causal network is a Bayesian network with additional constraints on
the meaning of the arrows.
 Graphical models are a broader class that includes Bayesian networks.

Bayesian networks
 A typical Bayesian
network, showing both
the topology and the
conditional probability
tables (CPTs).

End of Lesson
Uncertainty
Pt.1 Probabilistic Methods

Lesson04-Uncertainty - Pt. 1 Probabilistic Methods.pptx

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