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BAYESIAN NETWORKS Ivan Bratko Faculty of Computer and Information Sc. University of Ljubljana

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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?

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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

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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

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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

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EXAMPLE BAYES NET burglary lightning sensor alarm call

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PROBABILISTC DEPENDENCIES AND CAUSALITY Belief networks define probabilistic dependencies (and independencies) among the variables They may also reflect causality (burglar triggers sensor)

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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.

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TERMINOLOGY Bayes network = belief network = probabilistic network = causal network

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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

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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, 2 n - 1 probabilities - too many! Belief networks enable that this number can usually be reduced in practice

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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

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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

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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

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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

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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)

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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,...)

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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)

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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.

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SAFER FORMULATION OF INDEPENDENCE C is independent of C's nondescendants given C's parents (only) and not C's descendants.

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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)

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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

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BURGLARY EXAMPLE 10 numbers plus structure of network are equivalent to 2 5 - 1= 31 numbers required to specify complete probability distribution (without structure information).

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EXAMPLE QUERIES FOR BELIEF NETWORKS p( burglary | alarm) = ? p( burglary lightning) = ? p( burglary | alarm ~lightning) = ? p( alarm ~call | burglary) = ?

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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)

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

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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)

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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

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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

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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

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COMMENTS Complexity of reasoning in belief networks grows exponentially with the number of nodes. Substantial algorithmic improvements required for large networks for improved efficiency.

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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 V i, V j be two variables in the network

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d-SEPARATION Nodes V i and V j are conditionally independent given set E if E d-separates V i and V j E d-separates V i, V j if all (undirected) paths (V i,V j ) are “blocked” by E If E d-separates V i, V j, then V i and V j are conditionally independent, given E We write I(V i,V j | E) This means: p(V i,V j | E) = p(V i | E) * p(V j | E)

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BLOCKING A PATH A path between V i and V j is blocked by nodes E if there is a “blocking node” V b on the path. V b blocks the path if one of the following holds: V b in E and both arcs on path lead out of V b, or V b in E and one arc on path leads into V b and one out, or neither V b nor any descendant of V b is in E, and both arcs on path lead into V b

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CONDITION 1 V b is a common cause: V b V i V j

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CONDITION 2 V b is a “closer, more direct cause” of V j than V i is V i Vb V j

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CONDITION 3 V b is not a common consequence of V i, V j V i V j V b V b not in E V d V d not in E

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