Artificial Intelligence

UncertaintyChapter 13

Uncertainty

What shall an agent do when not all is crystal clear?

Different types of uncertainty effecting an agent:The state of the world?The effect of actions?

Uncertain knowledge of the world:Inputs missingLimited precision in the sensorsIncorrect model: action state due to the complexityA changing world

Uncertainty

Let action At = leave for airport t minutes before flightWill At get me there on time?

Problems:1) partial observability (road state, other drivers' plans,

etc.)2) noisy sensors (traffic reports)3) uncertainty in action outcomes (flat tire, etc.)4) immense complexity of modeling and predicting traffic

(A1440 might reasonably be said to get me there on time but I'd have

to stay overnight in the airport ...)

Rational Decisions

A rational decision must consider:– The relative importance of the sub-goals– Utility theory– The degree of belief that the sub-goals will be

achieved

– Probability theory

Decision theory = probability theory + utility theory : the agent is rational if and only if it chooses the action

that yields the highest expected utility, averaged over

all possible outcomes of the action”

Using FOL for (Medical) Diagnosis

p Symptom(p, Toothache) Disease(p, Cavity)Not correct...

p Symptom(p, Toothache) Disease(p, Cavity) Disease(p, GumDisease) Disease(p, WisdomTooth)

Not complete...

p Disease(p, Cavity) Symptom(p, Toothache)Not correct...

Handling Uncertain Knowledge

Problems using first-order logic for diagnosis:Laziness: Too much work to make complete rules. Too much work to use themTheoretical ignorance: Complete theories are rarePractical ignorance: We can’t run all tests anyway

Probability can be used to summarize the lazinessand ignorance !

Probability

Compare the following:1) First-order logic:

“The patient has a cavity”2) Probabilistic:

“The probability that the patient has a cavity is 0.8”

1) Is either valid or not, depending on the state of the world

2) Validity depends on the agents perception history, the evidence

Probability

Subjective or Bayesian probability:Probabilities relate propositions to one's own state of

knowledgee.g., P(A25 | no reported accidents) = 0.06

These are not claims of some probabilistic tendency in the current situation (but might be learned from past experience

of similar situations)

Probabilities of propositions change with new evidence:e.g., P(A25 | no reported accidents, 5 a.m.) = 0.15

(Analogous to logical entailment status KB ╞ )

Probability

Probabilities are either:

Prior probability (unconditional , “obetingad”)Before any evidence is obtained

Posterior probability (conditional , “betingad”) After evidence is obtained

Probability

Notation for unconditional probability for a proposition A: P(A)

Ex: P(Cavity)=0.2 means:“the degree of belief for “Cavity” given no extra evidence is 0.2”

Axioms for probabilities:

)()()()(.3

0)(,1)(.2

1)(0.1

BAPBPAPBAP

FalsePTrueP

AP

Probability

The axioms of probability constrain the possible assignments of probabilities to propositions. An

agent that violates the axioms will behave irrationally

in some circumstances

Random variable

A random variable has a domain of possible valuesEach value has a assigned probability between 0 and 1The values are :

Mutually exclusive (disjoint): (only one of them are true)Complete (there is always one that is true)

Example: The random variable Weather:P(Weather=Sunny) = 0.7P(Weather=Rain) = 0.2P(Weather=Cloudy) = 0.08P(Weather=Snow) = 0.02

Random Variable

The random variable Weather as a whole is said to have a

probability distribution which is a vector (in the discrete

case):P(Weather) = [0.7 0.2 0.08 0.02]

(Notice the bold P which is used to denote the prob.distribution)

Random variable - Example

Example - The random variable Season: P(Season = Spring) = 0.26 or shorter: P(Spring)=0.26

P(Season = Summer)= 0.20P(Season = Autumn) = 0.28P(Season = Winter) = 0.26

The random variable Season has a domain <Spring, Summer, Autumn, Winter>and a probability distribution:P(Season) = [0.26 0.20 0.38 0.26] The values in the domain are :

Mutually exclusive (disjoint): (only one of them are true)Complete (there is always one that is true)

Probability ModelBegin with a set - the sample spacee.g., 6 possible rolls of a die. is a sample point/possible world/atomic event

A probability space or probability model is a sample spacewith an assignment P() for every 0 P() 1 Σ P () = 1e.g., P(1) P(2) P(3) P(4) P(5) P(6) 1/6.

An event A is any subset of

P(A) = Σ{ A} P()

E.g., P(die roll < 4) = 1/6 + 1/6 + 1/6 = 1/2

The Joint Probability Distribution

Assume that an agent describing the world using therandom variables X1, X2, ..Xn.

The joint probability distribution (or ”joint”) assigns values for all combinations of values on X1, X2, ..Xn.

Notation: P(X1, X2, ..Xn) (i.e. P bold)

The Joint Probability Distribution

Joint probability distribution for a set of r.v.s gives the probability of

every atomic event on those r.v.s (i.e., every sample point)

P(Weather,Cavity) = a 4 x 2 matrix of values:

Every question about a domain can be answered by the joint distribution because every event is a sum of sample points

Weather = sunny

rain cloudy

snow

Cavity = true

0.144

0.02

0.016 0.02

Cavity = false

0.576

0.08

0.064 0.08

WeatherWeather Season Season SunSun RainRain CloudCloud SnowSnow

SpringSpring 0.070.07 0.030.03 0.100.10 0.060.06 0.260.26

P(Spring)P(Spring) SummerSummer 0.130.13 0.010.01 0.050.05 0.010.01 ++0.200.20

P(Summer)P(Summer) AutumnAutumn 0.050.05 0.050.05 0.150.15 0.030.03 ++0.280.28

P(Autumn)P(Autumn) WinterWinter 0.050.05 0.010.01 0.100.10 0.100.10 ++0.260.26

P(Winter)P(Winter)0.300.30 0.100.10 0.400.40 0.200.20P(Sun)+P(Rain)+P(Cloud)+P(Snow) P(Sun)+P(Rain)+P(Cloud)+P(Snow) ==

1.001.00

ExExampleample:: P(Weather=Sun P(Weather=Sun Season=Summer) = 0.13 Season=Summer) = 0.13

Example of ”Joint” P(Season, Weather)

Conditional ProbabilityThe Posterior prob. (conditional prob.) after obtaining evidence:Notation: P( A|B ) means: ”The probability of A given that all we know is B”. Example:P( Sunny | Summer ) = 0.65

Is defined as: if P(B) 0

Can be rewritten as the product rule:

P(A B) = P(A|B)P(B) = P(B|A)P(A)

”For A and B to be true, B has to be true, and A has to be true given B”

)()(

)|(BPBAP

BAP

Conditional Probability

For the entire random variables:

should be interpreted as a set of equations for allpossible values on the random variables A and B.Example:

)|()(),( BABBA PPP

)|()(),( SeasonWeatherSeasonSeasonWeather PPP

Conditional Probability

A general version holds for whole distributions, e.g.,P(Weather,Cavity) = P(Weather|Cavity) P(Cavity)

(View as a 4 x 2 set of equations)

Chain rule is derived by successive application of product

rule:P(X1,...,Xn) = P(X1,...,Xn-1) P(Xn | X1,...,Xn-1)

= P(X1,...,Xn-2)P(Xn-1 | X1,...,Xn-2) P(Xn | X1,...,Xn-1)

= ... = n

i=1 P(Xi | X1,...,Xi-1)

Bayes’ Rule

The left side of the product rule is symmetric w.r.t B and A:

Equating the two right-hand sides yields Bayes’ rule:

)()()|(

)|(AP

BPBAPABP

)|()()(

)|()()(

BAPBPBAP

ABPAPBAP

Bayes' Rule

Useful for assessing diagnostic probability from causal probability:

P(Cause|Effect) = P(Effect|

Cause)P(Cause)

P(Effect)

Example of Medical Diagnosis using Bayes’ rule

Known facts:Meningitis causes stiff neck 50% of the time.The probability of a patient having meningitis (M) is

1/50.000.The probability of a patient having stiff neck (S) is 1/20.Question:What is the probability of meningitis given stiff neck ?Solution:P(S|M)=0.5P(M) = 1/50.000 Note: posterior probability of

P(S) = 1/20 meningitis still very small!!0002.0

20/1

50000/15.0

P(S)

P(M)M)|P(SS)|P(M

Bayes' Rule

In distribution form

P(Y|X) = = P(X|Y) P(Y)P(X|Y) P(Y)

P(X)

Combining Evidence

Task: Compute P(Cavity|Toothache Catch )

1. Rewrite using the definition and use the joint. With N evidence variables, the “joint” will be an N dimensional table. It is often impossible to compute probabilities for all entries in the table.

2. Rewrite using Bayes’ rule. This also requires a lot of cond.prob. to be estimated. Other methods are to prefer.

Bayes' Rule and Conditional Independence

P(Cavity|toothache catch) = P(toothache catch|Cavity) P(Cavity) = P(toothache|Cavity) P(catch|Cavity) P(Cavity)

This is an example of a naive Bayes model:

P(Cause,Effect1,...,Effectn) = P(Cause) i P(Effecti|Cause)

Total number of parameters is linear in n

Summary

Probability can be used to reason about uncertainty 

Uncertainty arises because of both laziness and ignorance and it is inescapable

in complex, dynamic, or inaccessible worlds. 

Probabilities summarize the agent's beliefs.

Summary

Basic probability statements include prior probabilities and conditional probabilities over simple and complex propositions

The full joint probability distribution specifies the probability of

each complete as assignment of values to random variables. It is

usually too large to create or use in its explicit form

The axioms of probability constrain the possible assignments of probabilities to propositions. An agent that violates the axioms

will behave irrationally in some circumstances

Summary

Bayes' rule allows unknown probabilities to be computed from known conditional probabilities, usually in the causal direction.

With many pieces of evidence it will in general run into the same scaling problems as does the full joint distribution

Conditional independence brought about by direct causal relationships in the domain might allow the full joint distribution to

be factored into smaller, conditional distributions

The naive Bayes model assumes the conditional independence of all effect variables, given a single cause variable, and grows linearly with the number of effects

Next!

Bayesian Network!Chapter 14

Artificial Intelligence

Bayesian NetworkChapter 14

Introduction

Most application requires a way of handling uncertainty to be able to make adequate decisions

Bayesian decision theory is a fundamental statistical approach to the problem of complex decision making

The expert system developed in the middle of the 1970s used probabilistic techniques

Promising results - but not sufficient because of the exponential number of probabilities required in the full joint distribution

Bayesian Network - Syntax

A Bayesian network (or belief network) holds certain properties:

1. A set of random variables U={A1,…,An}, where each variable has a finite set of mutually exclusive states: Aj={a1,…, am}

2. A set of directed links or arrow connects pairs of nodes {A1 An}

3. Each node has a conditional probability table (CPT) that quantifies the effect that the parents have on the node

4. The graph is a directed acyclic graph (DAG)

Example

Topology of network encodes conditional independence assertions:

Weather is independent of the other variables

Toothache and Catch are conditionally independent given Cavity

Example

I'm at work, neighbor John calls to say my alarm is ringing, but

neighbor Mary doesn't call. Sometimes it is set off by minorearthquakes. Is there a burglar?

Variables: Burglar, Earthquake, Alarm, JohnCalls, MaryCallsNetwork topology reflects “causal” knowledge:

A burglar can set the alarm offAn earthquake can set the alarm offThe alarm can cause Mary to callThe alarm can cause John to call

Example

Bayesian Network - Semantics

Every Bayesian network provides a complete description of the

domain and has a joint probability distribution:

In order to construct a Bayesian network with the correct structure for the domain, we need to choose parents for each node such that this property holds

The parents of node Xi should contain all those nodes in

X1,…,Xi-1 that directly influence Xi

.))(|(),...,(1

1

n

iiin XParentsXPXXP

Example - Semantics

"Global" semantics defines the full joint distribution as

the product of the local conditional distributions:

e.g., P(j m a b e) = P(j|a) P(m|a) P(a|b,e) P(b) P(e)

Bayesian Network - Complexity

Since a Bayesian network is a complete and nonredundant representation of the domain it is often more compact than the full joint

General property of locally structure systems

In a locally structured, each subcomponent interacts directly with only a bounded number of other components

It is reasonable to suppose that in most domains each random variable is directly influenced by at most k others for some constant k

Bayesian network - Complexity

Consider Boolean variable for simplicity:

then the amount of information need to specify the CPT for a node will be at most 2k numbers, so the complete network can be specified by n2k numbers. The full joint contains 2n

For example, suppose we have 20 nodes (n=20) and each

node has 5 parents (k=5). Then the Bayesian network requires 640 numbers, but the full joint will require over a million

Bayesian network - Complexity

I.e., grows linearly with n, vs. O(2n) for the full joint distribution

For burglary net, 1 + 1 + 4 + 2 + 2 = 10 numbers (vs. 25-1 = 31)

Example

Conditional Independence

It is important to be aware of that the links between the variables represent a direct causal relationship

New information can be “transmitted” trough nodes in the

same and opposite directions of the links

This effects the properties of conditional dependency and

independency in Bayesian belief networks

Local Semantics

Local semantics: each node is conditionally independentof its nondescendants given its parents

Theorem: Local semantics global semantics

Constructing Bayesian Networks

Need a method such that a series of locally testable assertions ofconditional independence guarantees the required global semantics

Choose the set of relevant variables that describe the domainChoose an ordering of variables X1,...,Xn

For i = 1 to nadd Xi to the network

select parents from X1,...,Xi-1 such thatP(Xi|Parents(Xi)) = P(Xi|X1,..., Xi-1)

This choice of parents guarantees the global semantics:

P(X1,...,Xn) = ni = 1 P(Xi | X1,..., Xi-1) (chain rule)

= ni = 1 P(Xi|Parents(Xi))(by construction)

Learning with BN - Problems

Steps are often intermingled in practice

Judgments of conditional independence and/or cause and effect

can influence problem formulation

Assessments in probability may lead to changes in the network

structure

Learning with BN

Learning with BN

Learning with BN

Learning with BN

Inference

The basic task for any probabilistic inference system is to compute the posterior probability distribution for a set of query variables, given exact values for some evidence variables

Bayesian network are flexible enough so that any node can

serve as either a query or an evidence variable

By adding the evidence we get the posterior probability distribution our query variable will give

Inference

The task of interest is to compute: P(X|E)

Summary

Bayesian networks, a well-developed representation for uncertain knowledge

A Bayesian network is a directed acyclic graph whose nodes correspond to random variables; each node has a conditional distribution for the node, given its parents

Bayesian networks provide a concise way to represent conditional independence relationships in the domain

Summary

A Bayesian network specifies a full joint distribution; Each joint entry is defined as the product of the corresponding entries in the local conditional distributions

A Bayesian network is often exponentially smaller than the full joint distribution

Learning in Bayesian networks involves: Parametric estimation, parametric optimization, learning structures, parametric optimization combined with learning

structures

Next Time!

Repetition!