Bayesian Networks

Chapter 2 (Duda et al.) – Section 2.11

CS479/679 Pattern RecognitionDr. George Bebis

Statistical Dependences Between Variables

•Many times, the only knowledge we have about a distribution is which variables are or are not dependent.

•Such dependencies can be represented efficiently using a Bayesian Network (or Belief Networks).

Example of Dependencies

• State of an automobile– Engine temperature– Brake fluid pressure– Tire air pressure– Wire voltages

• Causally related variables– Engine temperature– Coolant temperature

• NOT causally related variables– Engine oil pressure– Tire air pressure

Bayesian Net Applications

• Microsoft (Answer Wizard, Print Troubleshooter)

• US Army: SAIP (Battalion Detection from SAR, IR etc.)

• NASA: Vista (DSS for Space Shuttle)

• GE: Gems (real-time monitoring of utility generators)

Definitions and Notation

• A bayesian net is usually a Directed Acyclic Graph (DAG)• Each node represents a system variable.• Each variable assumes certain states (i.e., values).

Relationships Between Nodes

• A link joining two nodes is directional and represents a causal influence (e.g., X depends on A or A influences X)

• Influences could be direct or indirect (e.g., A influences X directly and A influences C indirectly through X).

Parent/Children Nodes

• Parent nodes P of X– the nodes directly before X (connected to X)

• Children nodes C of X:– the nodes directly after X (X is connected to them)

Prior / Conditional Probabilities

• Each variable is associated with prior or conditional probabilities (discrete or continuous) .

probabilities

sum to 1

Markov Property

“Each node is conditionally independent of its ancestors given its parents”

Computing Joint Probabilities

• Using the chain rule, the joint probability of a set of variables x1, x2, …, xn is given as:

• Using the Markov property (i.e., node xi is conditionally independent of its ancestors given its parents πi), we have :

1 2 2 3 1( / ,..., ) ( / ,..., )... ( / ) ( )n n n n np x x x p x x x p x x p x

=

much simpler!

1 2( , ,..., )np x x x

1 21

( , ,..., ) ( / )n

n i ii

p x x x p x

Computing Joint Probabilities (cont’d)

• We can compute the probability of any configuration of variables in the joint density, e.g.:

P(a3, b1, x2, c3, d2)=P(a3)P(b1)P(x2 /a3,b1)P(c3 /x2)P(d2 /x2)=

0.25 x 0.6 x 0.4 x 0.5 x 0.4 = 0.012

Computing the Probability at a Node

• e.g., determine the probability at D

Fundamental Problems in Bayesian Nets

• Evaluation (inference): Given the model and the values of the observed variables (evidence), estimate the values of some other nodes (typically hidden nodes).

• Learning: Given training data and prior information (e.g., expert knowledge, causal relationships), estimate the network structure, or the parameters of the distribution, or both.

Example: Medical Diagnosis

Uppermost nodes: biological agents (bacteria, virus)

Intermediate nodes: diseases

Lowermost nodes: symptoms

• Given some evidence (biological agents, symptoms), find most likely disease.

causes

effects

Evaluation (Inference) Problem

• In general, if X denotes the query variables and e denotes the evidence, then

where α=1/P(e) is a constant of proportionality.

( , )( / ) ( , )

( )

P eP e P e

P e

XX X

Evaluation (Inference) Problem (cont’d)

• Exact inference is an NP-hard problem because the number of terms in the summations (or integrals) for discrete (or continuous) variables grows exponentially with increasing number of variables.

• For some restricted classes of networks (e.g., singly connected networks where there is no more than one path between any two nodes) exact inference can be efficiently solved in time linear in the number of nodes.

Evaluation (Inference) Problem (cont’d)

• For singly connected Bayesian networks:

• However, approximate inference methods have to be used in most cases.

– Sampling (Monte Carlo) methods– Variational methods– Loopy belief propagation

( / ) ( / , ) ( / ) ( / )

: , :C P P C

C P

P e P e e P e P e

e childrennodes e parent nodes

X X X X

Example• Classify a fish given that the fish is light (c1) and was caught

in south Atlantic (b2) -- no evidence about what time of the year the fish was caught nor its thickness.

Example (cont’d)

( , )

( / ) ( , )( )

P eP e P e

P e

XX X

Example (cont’d)

Example (cont’d)

• Similarly, P(x2 / c1,b2)=α 0.066

• Normalize probabilities (not needed necessarily):

P(x1 /c1,b2)+ P(x2 /c1,b2)=1 (α=1/0.18)

P(x1 /c1,b2)= 0.73

P(x2 /c1,b2)= 0.27 salmon

Another Example

• You have a new burglar alarm installed at home.

• It is fairly reliable at detecting burglary, but also sometimes responds to minor earthquakes.

• You have two neighbors, Ali and Veli, who promised to call you at work when they hear the alarm.

Another Example (cont’d)

• Ali always calls when he hears the alarm, but sometimes confuses telephone ringing with the alarm and calls too.

• Veli likes loud music and sometimes misses the alarm.

Another Example (cont’d)

• Design a Bayesian network to estimate various probabilities– e.g., given the evidence of who has or has not called,

we would like to estimate the probability of a burglary.

Another Example (cont’d)

• What are the main variables?– Alarm– Causes • Burglary, Earthquake

– Effects• Ali calls, Veli calls

Another Example (cont’d)

• What are the conditional dependencies among them?– Burglary (B) and earthquake (E) directly affect the

probability of the alarm (A) going off– Whether or not Ali calls (AC) or Veli calls (VC)

depends only on the alarm.

Another Example (cont’d)

Another Example (cont’d)

• What is the probability that the alarm has sounded but neither a burglary nor an earthquake has occurred, and both Ali and Veli call?

Another Example (cont’d)• What is the probability that there is a burglary

given that Ali calls?

• What about if Veli also calls right after Ali hangs up?

Naïve Bayesian Network

• When dependency relationships among features are unknown, we can assume that features are conditionally independent given the class:

• Simple assumption but usually works well in practice!

Naïve Bayesian Network