reasoning with bayesian networksusers.cecs.anu.edu.au/~jinbo/pr/lecture01.pdf · lecture 1:...

Probability CalculusBayesian Networks

Reasoning with Bayesian NetworksLecture 1: Probability Calculus, Bayesian Networks

Jinbo Huang

NICTA and ANU

Jinbo Huang Reasoning with Bayesian Networks


Overview of the Course

I Probability calculus, Bayesian networks

I Inference by variable elimination, factor elimination,conditioning

I Models for graph decomposition

I Most likely instantiations

I Complexity of probabilistic inference

I Compiling Bayesian networks

I Inference with local structure

I Applications



ProbabilitiesUpdating BeliefsIndependenceProperties of BeliefsSoft Evidence

Probabilities of Worlds

Worlds modeled with propositional variables

A world is complete instantiation of variables

Probabilities of all worlds add up to 1

An event is a set of worlds




Probabilities of Events

Pr(α)def=

∑ω∈α

Pr(ω)

I Pr(Earthquake) = Pr(ω1) + Pr(ω2) + Pr(ω3) + Pr(ω4) = .1

I Pr(¬Burglary) = .8

I Pr(Alarm) = .2442




Propositional Sentences as Events

Pr((Earthquake ∨ Burglary) ∧ ¬Alarm) =?

Interpret world ω as model (satisfying assignment) forpropositional sentence α

Pr(α)def=

∑ω|=α

Pr(ω)




Updating Beliefs

Suppose we’re given evidence β that Alarm = true

Need to update Pr(.) to Pr(.|β)




Updating Beliefs

Pr(β|β) = 1, Pr(¬β|β) = 0

Partition worlds into those |= β and those |= ¬βI Pr(ω|β) = 0, for all ω |= ¬β

I Relative beliefs stay put, for ω |= β

I∑

ω|=β Pr(ω) = Pr(β)

I∑

ω|=β Pr(ω|β) = 1

I Above imply we normalize by constant Pr(β)

Pr(ω|β)def=

{0, if ω |= ¬βPr(ω)/Pr(β), if ω |= β




Updating Beliefs

Pr(β|β) = 1, Pr(¬β|β) = 0

Partition worlds into those |= β and those |= ¬βI Pr(ω|β) = 0, for all ω |= ¬βI Relative beliefs stay put, for ω |= β

I∑

ω|=β Pr(ω) = Pr(β)

I∑

ω|=β Pr(ω|β) = 1

I Above imply we normalize by constant Pr(β)

Pr(ω|β)def=

{0, if ω |= ¬βPr(ω)/Pr(β), if ω |= β




Updating Beliefs




Bayes Conditioning

Pr(α|β) =∑ω|=α

Pr(ω|β)

=∑

ω|=α, ω|=β

Pr(ω|β) +∑

ω|=α, ω|=¬β

Pr(ω|β)

=∑

ω|=α, ω|=β

Pr(ω|β) =∑

ω|=α∧β

Pr(ω|β)

=∑

ω|=α∧β

Pr(ω)/Pr(β) =1

Pr(β)

∑ω|=α∧β

Pr(ω)

=Pr(α ∧ β)

Pr(β)




Bayes Conditioning

Pr(Burglary) = .2

Pr(Burglary |Alarm) ≈ .741 ↑Pr(Burglary |Alarm ∧ Earthquake) ≈ .253 ↓Pr(Burglary |Alarm ∧ ¬Earthquake) ≈ .957 ↑




Bayes Conditioning

Pr(Burglary) = .2Pr(Burglary |Alarm) ≈ .741 ↑

Pr(Burglary |Alarm ∧ Earthquake) ≈ .253 ↓Pr(Burglary |Alarm ∧ ¬Earthquake) ≈ .957 ↑




Bayes Conditioning

Pr(Burglary) = .2Pr(Burglary |Alarm) ≈ .741 ↑Pr(Burglary |Alarm ∧ Earthquake) ≈ .253 ↓

Pr(Burglary |Alarm ∧ ¬Earthquake) ≈ .957 ↑




Bayes Conditioning

Pr(Burglary) = .2Pr(Burglary |Alarm) ≈ .741 ↑Pr(Burglary |Alarm ∧ Earthquake) ≈ .253 ↓Pr(Burglary |Alarm ∧ ¬Earthquake) ≈ .957 ↑




Independence

Pr(Earthquake) = .1Pr(Earthquake|Burglary) = .1

Event α independent of event β if

Pr(α|β) = Pr(α) or Pr(β) = 0

Or equivalentlyPr(α ∧ β) = Pr(α) Pr(β)

Independence is symmetric




Conditional Independence

Independence is a dynamic notion

Burglary independent of Earthquake, but not after acceptingevidence Alarm

Pr(Burglary |Alarm) ≈ .741Pr(Burglary |Alarm ∧ Earthquake) ≈ .253

α independent of β given γ if

Pr(α ∧ β|γ) = Pr(α|γ) Pr(β|γ) or Pr(γ) = 0




Variable Independence

For disjoint sets of variables X, Y, and Z

IPr(X, Z, Y) : X independent of Y given Z in Pr

Means x independent of y given z for all instantiations of x, y, z




Properties of Beliefs: Chain Rule

Pr(α1 ∧ α2 ∧ . . . ∧ αn)

= Pr(α1|α2 ∧ . . . ∧ αn) Pr(α2|α3 ∧ . . . ∧ αn) . . .Pr(αn)




Properties of Beliefs: Case Analysis

Pr(α) =n∑

i=1

Pr(α ∧ βi )

Or equivalently

Pr(α) =n∑

i=1

Pr(α|βi ) Pr(βi )

Where events βi are mutually exclusive and collectively exhaustive




Properties of Beliefs: Case Analysis

Special case of n = 2

Pr(α) = Pr(α ∧ β) + Pr(α ∧ ¬β)

Or equivalently

Pr(α) = Pr(α|β) Pr(β) + Pr(α|¬β) Pr(¬β)




Properties of Beliefs: Bayes Rule

Pr(α|β) =Pr(β|α) Pr(α)

Pr(β)





Example: patient tested positive for disease

I D: patient has disease

I T : test comes out positive

I Would like to know Pr(D|T )

We know

I Pr(D) = 1/1000

I Pr(T |¬D) = 2/100 (false positive)

I Pr(¬T |D) = 5/100 (false negative)

Using Bayes rule: Pr(D|T ) = Pr(T |D) Pr(D)Pr(T )





Computing Pr(T ): case analysis

Pr(T ) = Pr(T |D) Pr(D) + Pr(T |¬D) Pr(¬D)

=95

100× 1

1000+

2

200× 999

1000=

2093

100000

Pr(D|T ) =95

2093≈ 4.5%




Soft Evidence

So far we’ve considered hard evidence, that some event hasoccurred

Soft evidence: neighbor with hearing problem calls to tell us theyheard alarm

May not confirm Alarm, but can still increase our belief in Alarm

Two main methods to specify strength of soft evidence




Soft Evidence: All Things Considered

“After receiving my neighbor’s call, my belief in Alarm is now .85”

Soft evidence as constraint Pr′(β) = q

Not a statement about strength of evidence per se, but result of itsintegration with our initial beliefs

Scale beliefs in worlds |= β so they add up to qScale beliefs in worlds |= ¬β so they add up to 1− q




Soft Evidence: All Things Considered

For worlds

Pr′(ω)def=

{q

Pr(β) Pr(ω), if ω |= β1−q

Pr(¬β) Pr(ω), if ω |= ¬β

For events (Jeffery’s rule)

Pr′(α) = q Pr(α|β) + (1− q) Pr(α|¬β)

Bayes conditioning is special case of q = 1




Soft Evidence: Nothing Else Considered

Would like to declare strength of evidence independently of currentbeliefs

Need notion of odds of event β: O(β)def= Pr(β)

Pr(¬β)

Declare evidence β by Bayes factor k = O′(β)O(β)

“My neighbor’s call increases odds of Alarm by factor of 4”

Compute Pr′(β) and fall back on Jeffrey’s rule to update beliefs



Capturing Independence GraphicallyConditional Probability TablesBayesian NetworksGraphoid Axiomsd-separation

Probabilistic Reasoning

Basic task: belief updating in face of hard and soft evidence

Can be done in principle using definitions we’ve seen

Inefficient, requires joint probability tables (exponential)

Also a modeling difficulty

I Specifying joint probabilities is tedious

I Beliefs (e.g., independencies) held by human experts noteasily enforced




Capturing Independence Graphically

Evidence R couldincrease Pr(A),Pr(C )

But not if we know¬A

C independent of Rgiven ¬A





Parents(V): variables withedge to V

Descendants(V): variablesto which ∃ directed pathfrom V

Nondescendants(V): allexcept V, Parents(V), andDescendants(V)




Markovian Assumptions

Every variable is conditionally independent of its nondescendantsgiven its parents

I (V ,Parents(V ),Nondescendants(V ))

Given direct causes of variable, our beliefs in that variable are nolonger influenced by any other variable except possibly by its effects





I (C ,A, {B,E ,R})

I (R,E , {A,B,C})

I (A, {B,E},R)

I (B, ∅, {E ,R})

I (E , ∅,B)




Conditional Probability Tables

DAG G together with Markov(G ) declares a set of independencies,but does not define Pr

Pr is uniquely defined if a conditional probability table (CPT) isprovided for each variable X

CPT: Pr(x |u) for every value x of variable X and everyinstantiation u of parents U





Pr(c|a)

Pr(r |e)

Pr(a|b, e)

Pr(e)

Pr(b)





Half of probabilitiesare redundant

Only need 10probabilities forwhole DAG




Bayesian Network

A pair (G ,Θ) where

I G is DAG over variables Z, called network structure

I Θ is set of CPTs, one for each variable in Z, called networkparametrization




Bayesian Network

ΘX |U: CPT for X andparents U

XU: network family

θx |u = Pr(x |u): networkparameter∑

x θx |u = 1 for everyparent instantiation u




Bayesian Network

Independence constraints imposed by network structure (DAG) andnumeric constraints imposed by network parametrization (CPTs)are satisfied by unique Pr

Pr(z)def=

∏θx|u∼z

θx |u (chain rule)

Bayesian network is implicit representation of this probabilitydistribution




Chain Rule Example

Pr(z)def=

∏θx|u∼z

θx |u (chain rule)

Pr(a, b, c, d , e) = θa θb|a θc|a θd |b,c θe|c

= (.6)(.2)(.2)(.9)(.1)

= .0216




Compactness of Bayesian Networks

Let d be max variable domain size, k max number of parents

Size of any CPT = O(dk+1)

Total number of network parameters = O(n · dk+1) for n-variablenetwork

Compact as long as k is relatively small, compared with jointprobability table (up to dn entries)




Properties of Probabilistic Independence

Distribution Pr represented by Bayesian network (G ,Θ) isguaranteed to satisfy every independence assumption in Markov(G )

IPr(X ,Parents(X ),Nondescendants(X ))

It may satisfy more

Derive independence by graphoid axioms




Graphoid Axioms: Symmetry

IPr(X,Z,Y)↔ IPr(Y,Z,X)

If learning y does not influence our belief in x, then learning x doesnot influence our belief in y either




Graphoid Axioms: Decomposition

IPr(X,Z,Y ∪W)→ IPr(X,Z,Y) ∧ IPr(X,Z,W)

If learning yw does not influence our belief in x, then learning yalone, or w alone, does not influence our belief in x either

If some information is irrelevant, then any part of it is alsoirrelevant

Reverse does not hold in general: Two pieces of information mayeach be irrelevant on their own, yet their combination may berelevant




Graphoid Axioms: Decomposition

IPr(X,Z,Y ∪W)→ IPr(X,Z,Y) ∧ IPr(X,Z,W)

In particular, can strengthen Markov(G )

IPr(X ,Parents(X ),W) for W ⊆ Nondescendants(X )

Useful for proving chain rule for Bayesian networks from chain ruleof probability calculus




Graphoid Axioms: Weak Union

IPr(X,Z,Y ∪W)→ IPr(X,Z ∪ Y,W)

If information yw is not relevant to our belief in x, then partialinformation y will not make rest of information, w, relevant

In particular, can strengthen Markov(G )

IPr(X ,Parents(X ) ∪W,Nondescendants(X )\W)

for W ⊆ Nondescendants(X )




Graphoid Axioms: Contraction

IPr(X,Z,Y) ∧ IPr(X,Z ∪ Y,W)→ IPr(X,Z,Y ∪W)

If, after learning irrelevant information y, information w is found tobe irrelevant to our belief in x, then combined information ywmust have been irrelevant to start with




Intersection Axiom

IPr(X,Z ∪W,Y) ∧ IPr(X,Z ∪ Y,W)→ IPr(X,Z,Y ∪W)

for strictly positive distribution Pr

If information w is irrelevant given y, and information y is irrelevantgiven w, then their combination is irrelevant to start with

Not true in general, when there’re zero probabilities (determinism)




Graphical Test of Independence

Graphoid axioms produce independencies beyond Markov(G )

Derivation of independencies is nontrivial; need efficient way

Linear-time graphical test : d-separation




d-separation

dsepG (X,Z,Y): every path between X & Y is blocked by Z

dsepG (X,Z,Y)→ IPr(X,Z,Y) for every Pr induced by G(soundness)

Also enjoys weak completeness (discussed later)




d-separation: Paths and Valves




d-separation: Path Blocking

Path is blocked if any valve is closed

Sequential valve →W → closed iff W ∈ Z

Divergent valve ←W → closed iff W ∈ Z

Convergent value →W ← closed iff W 6∈ Z and none ofDescendants(W ) appears in Z




d-separation: Examples




d-separation: Complexity

Need not enumerate paths between X & Y

dsepG (X,Z,Y) iff X and Y are disconnected in new DAG G ′

obtained by pruning DAG G as follows

I Delete all leaves W 6∈ X ∪ Y ∪ Z

I Delete all edges outgoing from Z

Linear in size of G




d-separation: Soundness

dsepG (X,Z,Y)→ IPr(X,Z,Y) for Pr induced by G

Prove by showing every independence claimed by d-separation canbe derived from graphoid axioms




d-separation: Completeness?

dsepG (X,Z,Y)← IPr(X,Z,Y)?

Does not hold

I Consider X → Y → Z

I X not d-separated from Z (given ∅)I However, if θy |x = θy |x then X & Y are independent, so are

X & Z

Not surprising as d-separation is based on structure alone




d-separation: Weak Completeness

For every DAG G , ∃ parametrization Θ such that

dsepG (X,Z,Y)↔ IPr(X,Z,Y)

where Pr is induced by Bayesian network (G ,Θ)

Implies that one cannot improve on d-separation




Summary

Probabilities of worlds & events, Bayes conditioning, independence,chain rule, case analysis, Bayes rule, two methods for soft evidence

Capturing independence graphically, Markovian assumptions,conditional probabilities tables, Bayesian networks, chain rule,graphoid axioms, d-separation


reasoning with bayesian networksusers.cecs.anu.edu.au/~jinbo/pr/lecture01.pdf · lecture 1:...

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