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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

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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
  • 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
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 example4
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
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
EXAMPLE BAYES NET

burglary lightning

sensor

alarm call

probabilistc dependencies and causality
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
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
TERMINOLOGY

Bayes network =

belief network =

probabilistic network =

causal network

bayes networks definition
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
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
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
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
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
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
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)

slide17
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
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 care19
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
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
STATING PROBABILITIES IN BELIEF NETS

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

p( X | Y1Y2 ...)

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 example22
BURGLARY EXAMPLE

p(burglary) = 0.001

p(lightning) = 0.02

p(sensor | burglary  lightning) = 0.9

p(sensor | burglary  ~lightning) = 0.9

p(sensor | ~burglary  lightning) = 0.1

p(sensor | ~burglary  ~lightning) = 0.001

p(alarm | sensor) = 0.95

p(alarm | ~sensor) = 0.001

p(call | sensor) = 0.9

p(call | ~sensor) = 0.0

burglary example23
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
EXAMPLE QUERIES FOR BELIEF NETWORKS
  • p( burglary | alarm) = ?
  • p( burglary  lightning) = ?
  • p( burglary | alarm  ~lightning) = ?
  • p( alarm  ~call | burglary) = ?
probabilistic reasoning in belief nets
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
BAYES' FORMULA

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

reasoning rules
REASONING RULES

1. Probability of conjunction:

p( X1  X2 | Cond) = p( X1 | Cond) * p( X2 | X1  Cond)

2. Probability of a certain event:

p( X | Y1  ...  X  ...) = 1

3. Probability of impossible event:

p( X | Y1  ...  ~X  ...) = 0

4. Probability of negation:

p( ~X | Cond) = 1 – p( X | Cond)

slide28
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|XCond) / 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

a simple implementation in prolog
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

slide30
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
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
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 separation33
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
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
CONDITION 1

Vb is a common cause:

Vb

Vi Vj

condition 2
CONDITION 2
  • Vb is a “closer, more direct cause” of Vj than Vi is

Vi

Vb

Vj

condition 3
CONDITION 3
  • Vb is not a common consequence of Vi, Vj

Vi Vj

Vb Vb not in E

Vd Vd not in E