Bayesian Networks

Chapter 2 (Duda et al.) – Section 2.11

CS479/679 Pattern RecognitionDr. George Bebis

Statistical Dependence Between Variables

– Representing high-dimensional densities is very challenging since we need to estimate many parameters (e.g., kn)

• 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 Bayesian Networks (or Belief Networks).

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

Example of Dependencies

• Represent the 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 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., A influences X or X depends on A)

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

Prior / Conditional Probabilities

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

Markov Property

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

1 2 1 1( / ,..., ) ( / )np x x x p x Example:

1 : parents of x1

Computing Joint ProbabilitiesUsing the Markov property

• 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

Example

• We can compute the probability of any configuration of states 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

Fundamental Problems in Bayesian Nets

• Evaluation (inference): Given the values of the observed variables (evidence), estimate the values of the non-observed variables.

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

Inference Example: Medical Diagnosis

Uppermost nodes: biological agents (bacteria, virus)

Intermediate nodes: diseases

Lowermost nodes: symptoms

• Goal: 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

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

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 the 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:

• Approximate inference methods are typically 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

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.

• Design a Bayesian network to estimate the probability of a burglary given some evidence.

Another Example (cont’d)

• What are the system 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 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 both Veli and Ali call?

Naïve Bayesian Network

• Assuming that features are conditionally independent, the conditional class density can be simplified as follows:

• Sometimes works well in practice despite the strong assumption behind it.

Naïve Bayesian Network: