bayesian networks ivan bratko faculty of computer and information sc. university of ljubljana

BAYESIAN NETWORKS

Ivan Bratko

Faculty of Computer and Information Sc.

University of Ljubljana

BAYESIAN NETWORKS

Bayesian networks, or belief networks: an approach to handling uncertainty in knowledge-based systems

Mathematically well-founded in probability theory, unlike many other, earlier approaches to representing uncertain knowledge

Type of problems intended for belief nets: given that some things are known to be true, how likely are some other events?

BURGLARY EXAMPLE

We have an alarm system to warn about burglary.

We have received an automatic alarm phone call; how likely it is that there actually was a burglary?

We cannot tell about burglary for sure, but characterize it probabilistically instead

BURGLARY EXAMPLE

There are a number of events involved:

burglary

sensor that may be triggered by burglar

lightning that may also trigger the sensor

alarm that may be triggered by sensor

call that may be triggered by sensor

BAYES NET REPRESENTATION

There are variables (e.g. burglary, alarm) that can take values (e.g. alarm = true, burglary = false).

There are probabilistic relations among variables, e.g.:

if burglary = true

then it is more likely that alarm = true

EXAMPLE BAYES NET

burglary lightning

sensor

alarm call

PROBABILISTC DEPENDENCIESAND CAUSALITY

Belief networks define probabilistic dependencies (and independencies) among the variables

They may also reflect causality (burglar triggers sensor)

EXAMPLE OF REASONING IN BELIEF NETWORK

In normal situation, burglary is not very likely.

We receive automatic warning call; since sensor causes

warning call, the probability of sensor being on

increases; since burglary is a cause for triggering the

sensor, the probability of burglary increases.

Then we learn there was a storm. Lightning may also

trigger sensor. Since lightning now also explains how

the call happened, the probability of burglary decreases.

TERMINOLOGY

Bayes network =

belief network =

probabilistic network =

causal network

BAYES NETWORKS, DEFINITION

Bayes net is a DAG (direct acyclic graph)

Nodes ~ random variables

Link X Y intuitively means:

“X has direct influence on Y”

For each node: conditional probability table quantifying effects of parent nodes

MAJOR PROBLEM IN HANDLING UNCERTAINTY

In general, with uncertainty, the problem is the handling

of dependencies between events.

In principle, this can be handled by specifying the

complete probability distribution over all possible

combinations of variable values.

However, this is impractical or impossible: for n binary

variables, 2n - 1 probabilities - too many!

Belief networks enable that this number can usually be

reduced in practice

BURGLARY DOMAIN

Five events: B, L, S, A, C

Complete probability distribution:

p( B L S A C) = ...

p( ~B L S A C) = ...

p( ~B ~L S A C) = ...

p( ~B L ~S A C) = ...

... Total: 32 probabilities

WHY BELIEF NETS BECAME SO POPULAR?

If some things are mutually independent then not all conditional probabilities are needed.

p(XY) = p(X) p(Y|X), p(Y|X) needed

If X and Y independent:

p(XY) = p(X) p(Y), p(Y|X) not needed!

Belief networks provide an elegant way of stating independences

EXAMPLE FROM J. PEARL

Burglary Earthquake

Alarm

John calls Mary calls

Burglary causes alarm Earthquake cause alarm When they hear alarm, neighbours John and Mary phone Occasionally John confuses phone ring for alarm Occasionally Mary fails to hear alarm

PROBABILITIES

P(B) = 0.001, P(E) = 0.002

A P(J | A) A P(M | A)

T 0.90 T 0.70

F 0.05 F 0.01

B E P(A | BE)

T T 0.95

T F 0.95

F T 0.29

F F 0.001

HOW ARE INDEPENDENCIES STATED IN BELIEF NETS

A

B

C

D

If C is known to be true, then prob. of D independent of A, B

p( D | A B C) = p( D | C)

A1, A2, ..... non-descendants of C

B1 B2 ... parents of C

C

D1, D2, ... descendants of C

C is independent of C's non-descendants given C's parents

p( C | A1, ..., B1, ..., D1, ...) = p( C | B1, ..., D1, ...)

INDEPENDENCE ON NONDESCENDANTS REQUIRES CARE

EXAMPLE

a

parent of c b

c e nondescendants of c

d f

descendant of c

By applying rule about nondescendants:

p(c|ab) = p(c|b)

Because: c independent of c's nondesc. a given c's parents (node b)

INDEPENDENCE ON NONDESCENDANTS REQUIRES CARE

But, for this Bayesian network:

p(c|bdf) p(c|bd)

Athough f is c's nondesc., it cannot be ignored:

knowing f, e becomes more likely;

e may also cause d, so when e becomes more likely, c becomes less likely.

Problem is that descendant d is given.

SAFER FORMULATION OF INDEPENDENCE

C is independent of C's nondescendants given

C's parents (only) and not C's descendants.

STATING PROBABILITIES IN BELIEF NETS

For each node X with parents Y1, Y2, ..., specify conditional probabilities of form:

p( X | Y1Y2 ...)for all possible states of Y1, Y2, ...

Y1 Y2

X

Specify:

p( X | Y1, Y2)

p( X | ~Y1, Y2)

p( X | Y1, ~Y2)

p( X | ~Y1, ~Y2)

BURGLARY EXAMPLE

10 numbers plus structure of network

are equivalent to

25 - 1= 31 numbers required to specify

complete probability distribution (without

structure information).

EXAMPLE QUERIES FOR BELIEF NETWORKS

p( burglary | alarm) = ? p( burglary lightning) = ? p( burglary | alarm ~lightning) = ? p( alarm ~call | burglary) = ?

Probabilistic reasoning in belief nets

Easy in forward direction, from ancestors to descendents, e.g.:

p( alarm | burglary lightning) = ?

In backward direction, from descendants to ancestors,

apply Bayes' formula

p( B | A) = p(B) * p(A | B) / p(A)

BAYES' FORMULA

A variant of Bayes' formula to reason about probability of hypothesis H given evidence E in presence of background knowledge B:

)(

)|()()|(

Yp

XYpXpYXp

)|(

)|()|()|(

BEp

BHEpBHpBEHp

5. If condition involves a descendant of X then use Bayes' theorem:

If Cond0 = Y Cond where Y is a descendant of X in belief net

then p(X|Cond0) = p(X|Cond) * p(Y|XCond) / p(Y|Cond)

6. Cases when condition Cond does not involve a descendant of X:

(a) If X has no parents then p(X|Cond) = p(X), p(X) given

(b) If X has parents Parents then

)(_)|()|()|(

ParentstatespossibleSCondSpSXpCondXp

A SIMPLE IMPLEMENTATION IN PROLOG

In: I. Bratko, Prolog Programming for Artificial Intelligence, Third edition, Pearson Education 2001(Chapter 15)

An interaction with this program:

?- prob( burglary, [call], P).

P = 0.232137

Now we learn there was a heavy storm, so:

?- prob( burglary, [call, lightning], P).

P = 0.00892857

Lightning explains call, so burglary seems less likely. However, if the weather was fine then burglary becomes more likely:

?- prob( burglary, [call,not lightning],P).

P = 0.473934

COMMENTS

Complexity of reasoning in belief networks grows exponentially with the number of nodes.

Substantial algorithmic improvements required for large networks for improved efficiency.

d-SEPARATION

Follows from basic independence assumption of Bayes

networks

d-separation = direction-dependent separation

Let E = set of “evidence nodes” (subset of variables in

Bayes network)

Let Vi, Vj be two variables in the network

d-SEPARATION

Nodes Vi and Vj are conditionally independent given set E if E

d-separates Vi and Vj

E d-separates Vi, Vj if all (undirected) paths (Vi,Vj) are “blocked”

by E

If E d-separates Vi, Vj, then Vi and Vj are conditionally

independent, given E

We write I(Vi,Vj | E)

This means: p(Vi,Vj | E) = p(Vi | E) * p(Vj | E)

BLOCKING A PATH

A path between Vi and Vj is blocked by nodes E if there is a

“blocking node” Vb on the path. Vb blocks the path if one of

the following holds:

Vb in E and both arcs on path lead out of Vb, or

Vb in E and one arc on path leads into Vb and one out, or

neither Vb nor any descendant of Vb is in E, and both arcs

on path lead into Vb

CONDITION 1

Vb is a common cause:

Vb

Vi Vj

CONDITION 2

Vb is a “closer, more direct cause” of Vj than Vi is

Vi

Vb

Vj

CONDITION 3

Vb is not a common consequence of Vi, Vj

Vi Vj

Vb Vb not in E

Vd Vd not in E

bayesian networks ivan bratko faculty of computer and information sc. university of ljubljana

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