Transcript bayesian networks

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 DEPENDENCIES
AND 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,
A
T
F
B
T
T
F
F
P(J | A)
0.90
0.05
E
T
F
T
F
P(A | BE)
0.95
0.95
0.29
0.001
P(E) = 0.002
A
T
F
P(M | A)
0.70
0.01
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, .....
B1
B2
non-descendants of C
...
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
d
nondescendants of c
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 | 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 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 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
p(Y | X )
p( X | Y )  p( X )
p(Y )
A variant of Bayes' formula to reason about probability
of hypothesis H given evidence E in presence of
background knowledge B:
p( E | H  B )
p( H | E  B )  p( H | B )
p( E | B )
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)
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
p( X | Cond )  S possible_ states( Parent) p( X | S ) p( S | Cond )
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