Bayes’ Nets II

IndependenceIndependence

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Bayes’ Nets• A Bayes’ net is an efficient encoding

of a probabilistic model

(joint distribution) of a domain

• Questions we can ask:

– Inference: given a fixed BN, what is P(X | e)?

– Representation: given a BN graph, what kinds of

distributions can it encode?

– Modeling: what BN is most appropriate for a given

domain?

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Bayes’ Net SemanticsLet’s formalize the semantics of a Bayes’ net

• A set of nodes, one per variable X

• A directed, acyclic graph

• A conditional distribution for each node

– A collection of distributions over X, one for

each combination of parents’ values

– CPT: conditional probability table

– Description of a noisy “causal” process

A Bayes net = Topology (graph) + Local Conditional Probabilities 3

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 P(X1, X2, ..., Xn)

• BNs: Huge space savings!

Also easier to elicit local CPTs from data

Also turns out to be faster to answer queries (coming)

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Building the (Entire) Joint• We can take a Bayes’ net and build any entry

from the full joint distribution it encodes

– Typically, there’s no reason to build ALL of it– Typically, there’s no reason to build ALL of it

– We build what we need on the fly

• To emphasize:

every BN over a domain implicitly defines a joint distribution

over that domain, specified by local probabilities and graph

structure

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Bayes’ Nets so far• We now know:

– What is a Bayes’ net?

– What joint distribution does a Bayes’ net encode?

• Now: properties of that joint distribution (independence)

– Key idea: conditional independence

– Last class: assembled BNs using an intuitive notion of

conditional independence as causality

– Today: formalize these ideas

– Main goal: answer queries about conditional

independence and influence

• Next: how to compute posterior probabilities quickly (inference)

Bayes’ Nets: Assumptions• Assumptions we are required to make to define the Bayes

net when given the graph:

P(xi | x1, ..., xi-1) = P(xi | parents(Xi))

Xi {X1, ..., Xi-1}\Parents(Xi) | Parents(Xi)

Explicit

Assumptions

Implicit

• Probability distributions that satisfy the above (“chain-rule

→Bayes net”) conditional independence assumptions

– Often guaranteed to have many more conditional independences

– Additional conditional independences can be read off the graph

• Important for modeling:

understand assumptions made when choosing a Bayes net graph7

Implicit

Assumptions.

Conditional Independence• Reminder: independence

– X and Y are independent if

– X and Y are conditionally independent given Z– X and Y are conditionally independent given Z

– (Conditional) independence is a property of a distribution

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Example

• Can extract conditional independence assumptions directly from

simplifications in chain rule:

– For the order X,Y,Z,W the chain rule is:

P(X, Y, Z,W) = P(X) P(Y|X) P(Z|X,Y) P(W|X,Y,Z)P(X, Y, Z,W) = P(X) P(Y|X) P(Z|X,Y) P(W|X,Y,Z)

– Bayes net:

P(X, Y, Z,W) = P(X) P(Y|X) P(Z|Y) P(W|Z)

– The explicit assumptions: Z X | Y , W X | Z , W Y | Z

• Additional implied conditional independence assumptions?

– X W | Y9

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:Necessarily Independent:

no matter which CPTs you

put in the graph, two

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)

• Note: they actually could be independent: how?

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put in the graph, two

variables will be

independent

D-Separation: outline• Study independence properties for triples

• Any complex example can be analyzed using these

three canonical cases.11

Y

X Z

1. Causal Chains• This configuration is a “causal chain”

– Bayes net:

– Is X independent of Z?

X: Low pressure

Y: Rain

Z: TrafficNot necessarily

– Is X independent of Z given Y?

– Evidence along the chain “blocks” the influence

that goes form X to Z 12

Yes

2. Common Cause• Another basic configuration:

two effects of the same cause

– Are X and Z independent?

Not necessarily.

– Are X and Z independent given Y?

Y: Alarm

X: John callsYes– Are X and Z independent given Y?

– Observing the cause blocks

influence between effects.13

X: John calls

Z: Mary calls

Yes

3. Common Effect• Last configuration:

two causes of one effect (v-structure)

– Are X and Z independent?

Bayes’ Net: P(X, Z, Y) = P(X) P(Z) P(Y|X,Z)

Chain rule: P(X, Z, Y) = P(X) P(Z|X) P(Y|X,Z)

→ P(Z|X) = P(Z)

X: Burglary

Z: Earthquake

Yes

→ P(Z|X) = P(Z)

– Are X and Z independent given Y?

Not necessarily: seeing Alarm puts the Burglary and the

Earthquake in competition as explanation.

– This is backwards from the other cases.

Observing an effect activates influence

between possible causes.

Z: Earthquake

Y: Alarm

If there is alarm, by knowing

there is a burglary, the

probability of earthquake

decreases. This is called

“Explaining-away”

Triplets: Recap

Y

X Z

• Definition: (X, Y, Z) is an ACTIVE TRIPLET given the

evidence if and only if X and Z are not necessarily independent

given the evidence.

• We established in previous slides that the red-framed triplets are

active, the others are not. 15

Y

The General Case• Definition: A (undirected) path (X0, X1, ..., Xn) in the graph is an

ACTIVE PATH given the evidence if and only if every triple (Xi-1,

Xi, Xi+1) is active.

• Fact: X and Z are not necessarily independent given the evidence

if and only if at least one path (X, ..., Z) is an active path.if and only if at least one path (X, ..., Z) is an active path.

• Equivalent fact: X and Z are guaranteed to be independent given

the evidence if and only if all paths (X, ..., Z) are inactive paths.

• � General solution to analyse independencies:

only need to analyze the graph, not the CPT entries!

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Reachability / D-Separation• Active triplets:

– Causal chain A → B → C where B

is unobserved (either direction)

– Common cause A ← B → C

where B is unobserved

– Common effect (aka v-structure)

A → B ← C where B or one of its

Active Triplets Inactive Triplets

descendents is observed

• A path is active if each triple is active

• Question: Are X and Y

conditionally independent

given evidence vars {Z}?

– Yes, if X and Y “D-separated” by Z

– Look for active paths from X to Y

– No active paths → D-separated → independent!

• All it takes to block a path is a single inactive triplet17

D-Separation• Given query

(only having BN structure, no CPTs)

Method:

• Shade all evidence nodes

• For all (undirected!) paths between Xi and Xj• For all (undirected!) paths between Xi and Xj

– Check whether path is active

• If path active return

• (If reaching this point, means all paths have been

checked and shown inactive)

return

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Example• Are the following statements true?

Yes

Not necessarily

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Not necessarily

Example

Yes

Yes

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Yes

Not necessarily

Not necessarily

Example• Variables:

– R: Raining

– T: Traffic

– D: Roof drips

– S: I’m sad

• Questions:

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Yes

Not necessarily

Not necessarily

Topology of Bayes’ Net and

Conditional Independence

• If X and Y are d-separated given {Z1, ..., Zk} then no matter

what CPT’s you choose for each variable in the BN, you will

always have that

X Y | {Z1, ..., Zk}X Y | {Z1, ..., Zk}

• If X and Y are not d-separated given {Z1, ..., Zk} then for most

choices of CPT’s there will be a dependence between X an Y

given {Z1, ..., Zk}, but for some special choices of CPT’s X and

Y will be conditionally independent given {Z1, ..., Zk}.

• One such special choice:

have all CPT’s be a uniform distribution 22

All Conditional Independences• Given a Bayes net structure,

– we can run d-separation for all pairs of variables given any

subset of other variables (lots of work)

– In this way, we build a complete list of conditional

independences that are necessarily true in the formindependences that are necessarily true in the form

– This list determines

the set of probability distributions that can be represented

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Topology Limits Distributions• Given some graph topology G,

only certain joint distributions can

be encoded by G.

• The graph structure guarantees

certain (conditional)

independencesindependences

• (There might be more independence)

• Adding arcs increases the set of

distributions, but has several costs

• Full conditioning can

encode any distribution 24

Adding Extra Arcs: Coins• Extra arcs don’t prevent representing independence, just allow

dependence as well

• Adding unneeded arcs isn’t

wrong, it’s just inefficient25

Changing Bayes’ Net Structure• The same joint distribution can be encoded in many

different Bayes’ nets

– Causal structure tends to be the simplest

• Analysis question: given some edges, what other edges

do you need to add?do you need to add?

– One answer: fully connect the graph

– Better answer: don’t make any false conditional independence

assumptions

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Example: Alternate Alarm

If we reverse the edges, we

make different conditional

independence assumptions

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To capture the same joint

distribution, we have to add

more edges to the graph

Summary• Bayes nets compactly encode joint distributions

• Guaranteed independencies of distributions can

be deduced from BN graph structure

• D-separation gives precise conditional• D-separation gives precise conditional

independence guarantees from graph alone

• A Bayes’ net’s joint distribution may have further

(conditional) independence that is not detectable

until you inspect its specific distribution

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