bayesnetwork

Bayesian Networks

CSC 371: Spring 2012

Today’s Lecture• Recap: Joint distribution, independence,

marginal independence, conditional independence

• Bayesian networks

• Reading:– Sections 14.1-14.4 in AIMA [Russel & Norvig]

Marginal Independence

• Intuitively: if X ╨ Y, then– learning that Y=y does not change your belief in X– and this is true for all values y that Y could take

• For example, weather is marginally independent of the result of a coin toss

3

Marginal Independence

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Conditional Independence

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• Intuitively: if X ╨ Y | Z, then– learning that Y=y does not change your belief in X

when we already know Z=z– and this is true for all values y that Y could take

and all values z that Z could take• For example,

ExamGrade ╨ AssignmentGrade | UnderstoodMaterial

Conditional Independence

“…probability theory is more fundamentally concerned with the structure of reasoning and causation than with numbers.”

Glenn Shafer and Judea PearlIntroduction to Readings in Uncertain Reasoning,Morgan Kaufmann, 1990

Bayesian Network Motivation• We want a representation and reasoning system that is

based on conditional (and marginal) independence– Compact yet expressive representation– Efficient reasoning procedures

• Bayesian (Belief) Networks are such a representation– Named after Thomas Bayes (ca. 1702 –1761)– Term coined in 1985 by Judea Pearl (1936 – )– Their invention changed the primary focus of AI from logic to

probability!

Thomas Bayes Judea Pearl8

Bayesian Networks: Intuition• A graphical representation for a joint probability

distribution– Nodes are random variables

• Can be assigned (observed) or unassigned (unobserved)– Arcs are interactions between nodes

• Encode conditional independence• An arrow from one variable to another indicates direct

influence• Directed arcs between nodes reflect dependence

– A compact specification of full joint distributions• Some informal examples:

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UnderstoodMaterial

Assignment Grade

ExamGrade Alarm

Smoking At Sensor

Fire

Example of a simple Bayesian network

A B

C

• Probability model has simple factored form

• Directed edges => direct dependence

• Absence of an edge => conditional independence

• Also known as belief networks, graphical models, causal networks

• Other formulations, e.g., undirected graphical models

p(A,B,C) = p(A)p(B) p(C|A,B)

Bayesian Networks: Definition

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Bayesian Networks: Definition

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• Discrete Bayesian networks:– Domain of each variable is finite– Conditional probability distribution is a conditional

probability table – We will assume this discrete case

• But everything we say about independence (marginal & conditional) carries over to the continuous case

Examples of 3-way Bayesian Networks

A CB Marginal Independence:p(A,B,C) = p(A) p(B) p(C)


A

CB

Conditionally independent effects:p(A,B,C) = p(B|A)p(C|A)p(A)

B and C are conditionally independentGiven A

e.g., A is a disease, and we model B and C as conditionally independentsymptoms given A


A B

C

Independent Causes:p(A,B,C) = p(C|A,B)p(A)p(B)

“Explaining away” effect:Given C, observing A makes B less likelye.g., earthquake/burglary/alarm example

A and B are (marginally) independent but become dependent once C is known


A CB Markov dependence:p(A,B,C) = p(C|B) p(B|A)p(A)

Example: Burglar Alarm• I have a burglar alarm that is sometimes set off by

minor earthquakes. My two neighbors, John and Mary, promised to call me at work if they hear the alarm– Example inference task: suppose Mary calls and John doesn’t

call. What is the probability of a burglary?

• What are the random variables? – Burglary, Earthquake, Alarm, JohnCalls, MaryCalls

Example 5 binary variables:

B = a burglary occurs at your house E = an earthquake occurs at your house A = the alarm goes off J = John calls to report the alarm M = Mary calls to report the alarm

What is P(B | M, J) ? (for example) We can use the full joint distribution to answer this question

Requires 25 = 32 probabilities Can we use prior domain knowledge to come up with a

Bayesian network that requires fewer probabilities? What are the direct influence relationships?

A burglary can set the alarm off An earthquake can set the alarm off The alarm can cause Mary to call The alarm can cause John to call

Example: Burglar Alarm

What are the model parameters?

Conditional probability distributions

• To specify the full joint distribution, we need to specify a conditional distribution for each node given its parents: P (X | Parents(X))

Z1 Z2 Zn

X

…

P (X | Z1, …, Zn)

Example: Burglar Alarm

The joint probability distribution• For each node Xi, we know P(Xi | Parents(Xi))

• How do we get the full joint distribution P(X1, …, Xn)?

• Using chain rule:

• For example, P(j, m, a, b, e)= P(b) P(e) P(a | b, e) P(j | a) P(m | a)

•

n

iii

n

iiin XParentsXPXXXPXXP

11111 )(|,,|),,(

Constructing a Bayesian Network: Step 1

• Order the variables in terms of causality (may be a partial order)

e.g., {E, B} -> {A} -> {J, M}

• P(J, M, A, E, B) = P(J, M | A, E, B) P(A| E, B) P(E, B)

~ P(J, M | A) P(A| E, B) P(E) P(B)

~ P(J | A) P(M | A) P(A| E, B) P(E) P(B)

These CI assumptions are reflected in the graph

structure of the Bayesian network

Constructing this Bayesian Network: Step 2

• P(J, M, A, E, B) = P(J | A) P(M | A) P(A | E, B) P(E) P(B)

• There are 3 conditional probability tables (CPDs) to be determined: P(J | A), P(M | A), P(A | E, B)

– Requiring 2 + 2 + 4 = 8 probabilities

• And 2 marginal probabilities P(E), P(B) -> 2 more probabilities

• 2 + 2 + 4 + 1 + 1 = 10 numbers (vs. 25-1 = 31)

• Where do these probabilities come from?– Expert knowledge– From data (relative frequency estimates)– Or a combination of both

Number of Probabilities in Bayesian Networks

• Consider n binary variables

• Unconstrained joint distribution requires O(2n) probabilities

• If we have a Bayesian network, with a maximum of k parents for any node, then we need O(n 2k) probabilities

• Example– Full unconstrained joint distribution

• n = 30: need 109 probabilities for full joint distribution– Bayesian network

• n = 30, k = 4: need 480 probabilities

Constructing Bayesian networks1. Choose an ordering of variables X1, … , Xn

2. For i = 1 to n

– add Xi to the network

– select parents from X1, … ,Xi-1 such thatP(Xi | Parents(Xi)) = P(Xi | X1, ... Xi-1)

• Suppose we choose the ordering M, J, A, B, E

P(J | M) = P(J)?

Example


P(J | M) = P(J)? No

Example

Example contd.

• Deciding conditional independence is hard in noncausal directions– The causal direction seems much more natural

• Network is less compact: 1 + 2 + 4 + 2 + 4 = 13 numbers needed

A more realistic Bayes Network: Car diagnosis

• Initial observation: car won’t start• Orange: “broken, so fix it” nodes• Green: testable evidence• Gray: “hidden variables” to ensure sparse structure, reduce

parameteres

The Bayesian Network from a different Variable Ordering

Given a graph, can we “read off” conditional independencies?

Are there wrong network structures?

• Some variable orderings yield more compact, some less compact structures– Compact ones are better– But all representations resulting from this process

are correct– One extreme: the fully connected network is

always correct but rarely the best choice

• How can a network structure be wrong?– If it misses directed edges that are required

Summary• Bayesian networks provide a natural

representation for (causally induced) conditional independence

• Topology + conditional probability tables• Generally easy for domain experts to

construct

Probabilistic inference

• A general scenario:– Query variables: X– Evidence (observed) variables: E = e – Unobserved variables: Y

• If we know the full joint distribution P(X , E, Y), how can we perform inference about X?

• Problems– Full joint distributions are too large– Marginalizing out Y may involve too many summation

terms

y

yeXe

eXeEX ),,(

)(

),()|( P

P

PP

Conclusions…• Full joint distributions are intractable to work

with– Conditional independence assumptions allow us to

model real-world phenomena with much simpler models

– Bayesian networks are a systematic way to construct parsimonious structured distributions

• How do we do inference (reasoning) in Bayesian networks?– Systematic algorithms exist– Complexity depends on the structure of the graph

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