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Bayesian Networks

Tamara BergCS 590-133 Artificial Intelligence

Many slides throughout the course adapted from Svetlana Lazebnik, Dan Klein, Stuart Russell, Andrew Moore, Percy Liang, Luke Zettlemoyer, Rob Pless, Killian Weinberger, Deva Ramanan

Announcements• Some students in the back are having trouble

hearing the lecture due to talking.

• Please respect your fellow students. If you have a question or comment relevant to the course please share with all of us. Otherwise, don’t talk during lecture.

• Also, if you are having trouble hearing in the back there are plenty of seats further forward.

Reminder

• HW3 was released 2/27– Written questions only (no programming)– Due Tuesday, 3/18, 11:59pm

From last class

Random Variables

Random variables

be a realization of Let

A random variable is some aspect of the world about which we (may) have uncertainty.

Random variables can be:Binary (e.g. {true,false}, {spam/ham}), Take on a discrete set of values

(e.g. {Spring, Summer, Fall, Winter}), Or be continuous (e.g. [0 1]).

Joint Probability Distribution

Random variables

Joint Probability Distribution:

be a realization of Let

Also written

Gives a real value for all possible assignments.

Queries

Joint Probability Distribution:

Also written

Given a joint distribution, we can reason about unobserved variables given observations (evidence):

Stuff you care about Stuff you already know

Main kinds of models• Undirected (also called Markov Random Fields)

- links express constraints between variables.

• Directed (also called Bayesian Networks) - have a notion of causality -- one can regard an arc from A to B as indicating that A "causes" B.

Syntax Directed Acyclic Graph (DAG) Nodes: random variables

Can be assigned (observed)or unassigned (unobserved)

Arcs: interactions An arrow from one variable to another indicates

direct influence Encode conditional independence

Weather is independent of the other variables Toothache and Catch are conditionally independent

given Cavity Must form a directed, acyclic graph

Weather Cavity

Toothache Catch

Bayes Nets

Directed Graph, G = (X,E)

Nodes

Edges

Each node is associated with a random variable

Example

Joint Distribution

By Chain Rule (using the usual arithmetic ordering)

Directed Graphical Models

Directed Graph, G = (X,E)

Nodes

Edges

Each node is associated with a random variable

Definition of joint probability in a graphical model:

where are the parents of

Example

Joint Probability:

Example

00

1

1

00

1

10

0

1

1

00

1

1

10

0

10 1

0

1

Size of a Bayes’ Net• How big is a joint distribution over N Boolean variables?

2N

• How big is an N-node net if nodes have up to k parents?

O(N * 2k+1)

• Both give you the power to calculate• BNs: Huge space savings!• Also easier to elicit local CPTs• Also turns out to be faster to answer queries

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The joint probability distribution

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

Independence in a BN• Important question about a BN:

– Are two nodes independent given certain evidence?– If yes, can prove using algebra (tedious in general)– If no, can prove with a counter example– Example:

– Question: are X and Z necessarily independent?• Answer: no. Example: low pressure causes rain, which

causes traffic.• X can influence Z, Z can influence X (via Y)• Addendum: they could be independent: how?

X Y Z

Causal Chains• This configuration is a “causal chain”

– Is Z independent of X given Y?

– Evidence along the chain “blocks” the influence

X Y Z

Yes!

X: Project due

Y: No office hours

Z: Students panic

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Common Cause• Another basic configuration: two

effects of the same cause– Are X and Z independent?

– Are X and Z independent given Y?

– Observing the cause blocks influence between effects.

X

Y

Z

Yes!

Y: Homework due

X: Full attendance

Z: Students sleepy

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Common Effect• Last configuration: two causes of

one effect (v-structures)– Are X and Z independent?

• Yes: the ballgame and the rain cause traffic, but they are not correlated

• Still need to prove they must be (try it!)– Are X and Z independent given Y?

• No: seeing traffic puts the rain and the ballgame in competition as explanation

– This is backwards from the other cases• Observing an effect activates influence

between possible causes.

X

Y

Z

X: Raining

Z: Ballgame

Y: Traffic

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The General Case• Any complex example can be

analyzed using these three canonical cases

• General question: in a given BN, are two variables independent (given evidence)?

• Solution: analyze the graph

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Causal Chain

Common Cause

(Unobserved)Common Effect

Bayes Ball

• Shade all observed nodes. Place balls at the starting node, let them bounce around according to some rules, and ask if any of the balls reach any of the goal node.

• We need to know what happens when a ball arrives at a node on its way to the goal node.

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Example

Yes

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R

T

B

T’

Bayesian decision making• Suppose the agent has to make decisions about

the value of an unobserved query variable X based on the values of an observed evidence variable E

• Inference problem: given some evidence E = e, what is P(X | e)?

• Learning problem: estimate the parameters of the probabilistic model P(X | E) given training samples {(x1,e1), …, (xn,en)}

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?

y

yeXeeXeEX ),,()(),()|( P

PPP

Inference• Inference: calculating some

useful quantity from a joint probability distribution

• Examples:– Posterior probability:

– Most likely explanation:

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B E

A

J M

Inference – computing conditional probabilities

Marginalization:Conditional Probabilities:

Inference by Enumeration• Given unlimited time, inference in BNs is easy• Recipe:

– State the marginal probabilities you need– Figure out ALL the atomic probabilities you need– Calculate and combine them

• Example:

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B E

A

J M

Example: Enumeration• In this simple method, we only need the BN to

synthesize the joint entries

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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

yeXeeXeEX ),,()(),()|( P

PPP

Inference by Enumeration?

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Variable Elimination• Why is inference by enumeration on a Bayes

Net inefficient?– You join up the whole joint distribution before you sum

out the hidden variables– You end up repeating a lot of work!

• Idea: interleave joining and marginalizing!– Called “Variable Elimination”– Choosing the order to eliminate variables that

minimizes work is NP-hard, but *anything* sensible is much faster than inference by enumeration

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General Variable Elimination• Query:

• Start with initial factors:– Local CPTs (but instantiated by evidence)

• While there are still hidden variables (not Q or evidence):– Pick a hidden variable H– Join all factors mentioning H– Eliminate (sum out) H

• Join all remaining factors and normalize

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Example: Variable elimination

Query: What is the probability that a student attends class, given that they pass the exam?

[based on slides taken from UMBC CMSC 671, 2005]

P(pr|at,st) at st0.9 T T0.5 T F0.7 F T0.1 F F

attend study

preparedfair

pass

P(at)=.8P(st)=.6

P(fa)=.9

P(pa|at,pre,fa) pr at fa0.9 T T T0.1 T T F0.7 T F T0.1 T F F0.7 F T T0.1 F T F0.2 F F T0.1 F F F

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Join study factors

attend study

preparedfair

pass

P(at)=.8P(st)=.6

P(fa)=.9

Original Joint Marginalprep study attend P(pr|at,st) P(st) P(pr,st|sm) P(pr|sm)

T T T 0.9 0.6 0.54 0.74T F T 0.5 0.4 0.2T T F 0.7 0.6 0.42 0.46T F F 0.1 0.4 0.04 F T T 0.1 0.6 0.06 0.26F F T 0.5 0.4 0.2 F T F 0.3 0.6 0.18 0.54F F F 0.9 0.4 0.36

P(pa|at,pre,fa) pr at fa0.9 T T T0.1 T T F0.7 T F T0.1 T F F0.7 F T T0.1 F T F0.2 F F T0.1 F F F

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Marginalize out study

attend

prepared,study

fair

pass

P(at)=.8

P(fa)=.9

Original Joint Marginalprep study attend P(pr|at,st) P(st) P(pr,st|at) P(pr|at)

T T T 0.9 0.6 0.54 0.74T F T 0.5 0.4 0.2T T F 0.7 0.6 0.42 0.46T F F 0.1 0.4 0.04 F T T 0.1 0.6 0.06 0.26F F T 0.5 0.4 0.2 F T F 0.3 0.6 0.18 0.54F F F 0.9 0.4 0.36

P(pa|at,pre,fa) pr at fa0.9 T T T0.1 T T F0.7 T F T0.1 T F F0.7 F T T0.1 F T F0.2 F F T0.1 F F F

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Remove “study”

attend

preparedfair

pass

P(at)=.8

P(fa)=.9

P(pr|at) pr at0.74 T T0.46 T F0.26 F T0.54 F F

P(pa|at,pre,fa) pr at fa0.9 T T T0.1 T T F0.7 T F T0.1 T F F0.7 F T T0.1 F T F0.2 F F T0.1 F F F

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Join factors “fair”

attend

preparedfair

pass

P(at)=.8

P(fa)=.9

P(pr|at) prep attend0.74 T T0.46 T F0.26 F T0.54 F F

Original Joint Marginal

pa pre attend fairP(pa|

at,pre,fa) P(fair)P(pa,fa|sm,pre)

P(pa|sm,pre)

t T T T 0.9 0.9 0.81 0.82t T T F 0.1 0.1 0.01 t T F T 0.7 0.9 0.63 0.64t T F F 0.1 0.1 0.01 t F T T 0.7 0.9 0.63 0.64t F T F 0.1 0.1 0.01 t F F T 0.2 0.9 0.18 0.19t F F F 0.1 0.1 0.01

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Marginalize out “fair”

attend

prepared

pass,fair

P(at)=.8

P(pr|at) prep attend0.74 T T0.46 T F0.26 F T0.54 F F

Original Joint Marginal

pa pre attend fair P(pa|at,pre,fa) P(fair) P(pa,fa|at,pre) P(pa|at,pre)T T T T 0.9 0.9 0.81 0.82T T T F 0.1 0.1 0.01 T T F T 0.7 0.9 0.63 0.64T T F F 0.1 0.1 0.01 T F T T 0.7 0.9 0.63 0.64T F T F 0.1 0.1 0.01 T F F T 0.2 0.9 0.18 0.19T F F F 0.1 0.1 0.01

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Marginalize out “fair”

attend

prepared

pass

P(at)=.8

P(pr|at) prep attend0.74 T T0.46 T F0.26 F T0.54 F F

P(pa|at,pre) pa pre attend0.82 t T T0.64 t T F0.64 t F T0.19 t F F

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Join factors “prepared”

attend

prepared

pass

P(at)=.8

Original Joint Marginalpa pre attend P(pa|at,pr) P(pr|at) P(pa,pr|sm) P(pa|sm)t T T 0.82 0.74 0.6068 0.7732t T F 0.64 0.46 0.2944 0.397t F T 0.64 0.26 0.1664 t F F 0.19 0.54 0.1026

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Join factors “prepared”

attend

pass,prepared

P(at)=.8

Original Joint Marginalpa pre attend P(pa|at,pr) P(pr|at) P(pa,pr|at) P(pa|at)t T T 0.82 0.74 0.6068 0.7732t T F 0.64 0.46 0.2944 0.397t F T 0.64 0.26 0.1664 t F F 0.19 0.54 0.1026

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Join factors “prepared”

attend

pass

P(at)=.8

P(pa|at) pa attend0.7732 t T0.397 t F

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Join factors

attend

pass

P(at)=.8

Original Joint Normalized:pa attend P(pa|at) P(at) P(pa,sm) P(at|pa)T T 0.7732 0.8 0.61856 0.89T F 0.397 0.2 0.0794 0.11

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Join factors

attend,pass

Original Joint Normalized:pa attend P(pa|at) P(at) P(pa,at) P(at|pa)T T 0.7732 0.8 0.61856 0.89T F 0.397 0.2 0.0794 0.11

Bayesian network inference: Big picture

• Exact inference is intractable– There exist techniques to speed up

computations, but worst-case complexity is still exponential except in some classes of networks

• Approximate inference – Sampling, variational methods, message

passing / belief propagation…

Approximate Inference

Sampling (particle based method)

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Approximate Inference

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Sampling – the basics ...• Scrooge McDuck gives you

an ancient coin. • He wants to know what is

P(H) • You have no homework,

and nothing good is on television – so you toss it 1 Million times.

• You obtain 700000x Heads, and 300000x Tails.

• What is P(H)?

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Sampling – the basics ...

• Exactly, P(H)=0.7• Why?

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Monte Carlo Method

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Who is more likely to win? Green or Purple?

What is the probability that green wins, P(G)?

Two ways to solve this:1. Compute the exact probability.2. Play 100,000 games and see

how many times green wins.

Approximate Inference• Simulation has a name: sampling

• Sampling is a hot topic in machine learning,and it’s really simple

• Basic idea:– Draw N samples from a sampling distribution S– Compute an approximate posterior probability– Show this converges to the true probability P

• Why sample?– Learning: get samples from a distribution you don’t know– Inference: getting a sample is faster than computing the right

answer (e.g. with variable elimination)55

S

A

F

Forward Sampling

Cloudy

Sprinkler Rain

WetGrass

Cloudy

Sprinkler Rain

WetGrass

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+c 0.5-c 0.5

+c+s 0.1

-s 0.9-c +s 0.5

-s 0.5

+c+r 0.8

-r 0.2-c +r 0.2

-r 0.8

+s

+r+w 0.99

-w 0.01

-r

+w 0.90

-w 0.10

-s +r +w 0.90-w 0.10

-r +w 0.01-w 0.99

Samples:

+c, -s, +r, +w-c, +s, -r, +w

…

Forward Sampling• This process generates samples with probability:

…i.e. the BN’s joint probability

• Let the number of samples of an event be

• Then

• I.e., the sampling procedure is consistent57

Example• We’ll get a bunch of samples from the BN:

+c, -s, +r, +w+c, +s, +r, +w-c, +s, +r, -w+c, -s, +r, +w-c, -s, -r, +w

• If we want to know P(W)– We have counts <+w:4, -w:1>– Normalize to get P(W) = <+w:0.8, -w:0.2>– This will get closer to the true distribution with more samples– Can estimate anything else, too– What about P(C| +w)? P(C| +r, +w)? P(C| -r, -w)?– Fast: can use fewer samples if less time (what’s the drawback?)

Cloudy

Sprinkler Rain

WetGrass

C

S R

W

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Rejection Sampling• Let’s say we want P(C)

– No point keeping all samples around– Just tally counts of C as we go

• Let’s say we want P(C| +s)– Same thing: tally C outcomes, but

ignore (reject) samples which don’t have S=+s

– This is called rejection sampling– It is also consistent for conditional

probabilities (i.e., correct in the limit)

+c, -s, +r, +w

+c, +s, +r, +w

-c, +s, +r, -w

+c, -s, +r, +w

-c, -s, -r, +w

Cloudy

Sprinkler Rain

WetGrass

C

S R

W

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Likelihood Weighting• Problem with rejection sampling:

– If evidence is unlikely, you reject a lot of samples– You don’t exploit your evidence as you sample– Consider P(B|+a)

• Idea: fix evidence variables and sample the rest

• Problem: sample distribution not consistent!• Solution: weight by probability of evidence given parents

Burglary Alarm

Burglary Alarm

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-b, -a -b, -a -b, -a -b, -a+b, +a

-b +a -b, +a -b, +a -b, +a+b, +a

Likelihood Weighting• Sampling distribution if z sampled and e fixed evidence

• Now, samples have weights

• Together, weighted sampling distribution is consistent

Cloudy

R

C

S

W

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Likelihood Weighting

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+c 0.5-c 0.5

+c+s 0.1

-s 0.9-c +s 0.5

-s 0.5

+c+r 0.8

-r 0.2-c +r 0.2

-r 0.8

+s

+r+w 0.99

-w 0.01

-r

+w 0.90

-w 0.10

-s +r +w 0.90-w 0.10

-r +w 0.01-w 0.99

Samples:

+c, +s, +r, +w…

Cloudy

Sprinkler Rain

WetGrass

Cloudy

Sprinkler Rain

WetGrass

Inference: Sum over weights that match query value Divide by total sample weight What is P(C|+w,+r)?

Likelihood Weighting Example

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Cloudy Rainy Sprinkler Wet Grass Weight0 1 1 1 0.4950 0 1 1 0.450 0 1 1 0.450 0 1 1 0.451 0 1 1 0.09

Likelihood Weighting• Likelihood weighting is good

– We have taken evidence into account as we generate the sample

– E.g. here, W’s value will get picked based on the evidence values of S, R

– More of our samples will reflect the state of the world suggested by the evidence

• Likelihood weighting doesn’t solve all our problems– Evidence influences the choice of

downstream variables, but not upstream ones (C isn’t more likely to get a value matching the evidence)

• We would like to consider evidence when we sample every variable 65

Cloudy

Rain

C

S R

W

Gibbs Sampling

1. Set all evidence E to e2. Do forward sampling to obtain x1,...,xn

3. Repeat:1. Pick any variable Xi uniformly at random.2. Resample xi’ from p(Xi | x1,..., xi-1, xi+1,..., xn)3. Set all other xj’=xj

4. The new sample is x1’,..., xn’

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Markov Blanket

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X

Markov blanket of X: 1. All parents of X2. All children of X3. All parents of children of X

(except X itself)

X is conditionally independent from all other variables in the BN, given all variables in the markov blanket (besides X).

Inference Algorithms• Exact algorithms

– Elimination algorithm– Sum-product algorithm– Junction tree algorithm

• Sampling algorithms– Importance sampling– Markov chain Monte Carlo

• Variational algorithms– Mean field methods– Sum-product algorithm and variations– Semidefinite relaxations

Summary

• Sampling can be your salvation• The dominating approach to inference in

BNs• Approaches:

– Forward (/Prior) Sampling– Rejection Sampling– Likelihood Weighted Sampling– Gibbs Sampling

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