Introduction to Bayesian Networks

Representation: What is the joint probability distribution

P(B, E, R, A, N) over five binary variables?

• How many states in total?

• How many free parameters do we need?

• How many table entries should we sum over to get P(R=r)?

Joint Probability DistributionJoint Probability Distribution

Joint Probability DistributionJoint Probability Distribution

• Full joint distribution is sufficient to represent the complete domain and to do any type of probabilistic inferences

• Problems: n - number of random variables, d – number of values

– Space complexity. A full joint distribution requires to remember O(dn) numbers

– Inference complexity. Computing queries requires O(dn) steps

– Acquisition problem. Who is going to define all the probabilities?

Joint Probability DistributionJoint Probability Distribution

),,,,( NAREBP

Joint Probability DistributionJoint Probability Distribution

),,,|(),,|(),|()|()(),,,,( AREBNPREBAPEBRPBEPBPNAREBP

Joint Probability DistributionJoint Probability Distribution

VS

Bayesian networksBayesian networks

• A Bayesian Network is a graph in which:– A set of random variables makes up the nodes in the network.

– A set of directed links or arrows connects pairs of nodes.

– Each node has a conditional probability table that quantifies the effects the parents have on the node.

– Directed, acyclic graph (DAG), i.e. no directed cycles.

A Bayesian network modeling the joint probability distribution P(B, E, R, A, N)

P(B) P(E)

P(R|E)P(A|B,E)

P(N|A)

Conditional probability tables (CPT):

Bayesian NetworksBayesian Networks

P(B) P(E)

P(N|A)

P(R|E)P(A|B,E)

Independences in BNsIndependences in BNs

• Three basic independence structures:

Independences in BNsIndependences in BNs

• Indirect cause: Burglary is independent of NeighborCall given Alarm

– P(N|A, B) = P(N|A)

– P(N, B|A) = P(N|A)P(B|A)

Independences in BNsIndependences in BNs

• Common cause: Alarm is independent of RadioAnnounce given Earthquake

– P(A|E, R) = P(A|E)

– P(A, R|E) = P(A|E)P(R|E)

Independences in BNsIndependences in BNs

• Common effect: – Burglary is independent of Earthquake when Alarm is not known

– Burglary and Earthquake become dependent given Alarm!!!

Path blockingPath blocking

• With linear substructure

• With common cause structure

• With common effect structure

X YC in Z

X YC in Z

X Y

C or any of its descendants not in Z

Independences in BNsIndependences in BNs

• Earthquake ┴ Burglary | NeighborCall?• Burglary ┴ NeighborCall ?• Burglary ┴ RadioAnnounce | Earthquake?• Burglary ┴ RadioAnnounce | NeighborCall?

Markov AssumptionMarkov Assumption

• Each variable is independent on its non-descendants, given its parents in Bayesian networks

• N ┴ B, E, R | A• R ┴ B, A, N | E• …

Full Joint Distribution in BNsFull Joint Distribution in BNs

),,,,( NRAEBP

)()(),|()|()|( BPEPEBAPERPANP

Chain RuleChain Rule

n

iiin XPaXPXXXP

121 ))(|(),...,,(

Types of Inference TasksTypes of Inference Tasks

Compute the probability or most likely state of a set of query variables given observed values of some other variables

• Belief updating

• Most probable explanation (MPE) queries

• Maximum A Posteriori (MAP) queries

Inference DirectionInference Direction

• Types of inference by reasoning direction

ExamplesExamples

• Diagnostic inferences: from effect to causes.

• Causal Inferences: from causes to effects.

• Intercausal Inferences:

• Mixed Inference:

Influence diagramInfluence diagram

• Add action nodes and utility nodes to Bayesian networks to enable rational decision making

• Algorithm:For each value of action node

compute expected value of utility node given action, evidenceReturn MEU action

• Idea: compute value of acquiring each possible piece of evidence– Can be done directly from influence diagram

• Example: buying oil drilling rightsTwo blocks A and B, exactly one has oil, worth kPrior probabilities 0.5 each, mutually exclusive

Current price of each block is k/2“Consultant” offers accurate survey of A. Fair price?

• Solution: compute expected value of information= expected value of best action given the information

minus expected value of best action without information

• Survey may say “oil in A” or “no oil in A”, prob. 0.5 each (given!)

= [0.5 * value of “buy A” given “oil in A”+ 0.5 * value of “buy B” given “no oil in A”]- 0

= (0.5 * k/2) + (0.5 * k/2) - 0 = k/2

Value of informationValue of information

Oil

Survey

Buy

U