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Reasoning with Graphical Models Slides Set 2: Rina Dechter - - PowerPoint PPT Presentation

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Reasoning with Graphical Models Slides Set 2: Rina Dechter Reading: Darwiche chapters 4 Pearl: chapter 3 slides2 COMPSCI 2020 Outline Basics of probability theory DAGS, Markov(G), Bayesian networks Graphoids: axioms of for

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Reasoning with Graphical Models

Slides Set 2:

Rina Dechter

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Reading: Darwiche chapters 4 Pearl: chapter 3

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Outline

 Basics of probability theory  DAGS, Markov(G), Bayesian networks  Graphoids: axioms of for inferring

conditional independence (CI)

 D-separation: Inferring CIs in graphs

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Outline

 Basics of probability theory  DAGS, Markov(G), Bayesian networks  Graphoids: axioms of for inferring

conditional independence (CI)

 Capturing CIs by graphs  D-separation: Inferring CIs in graphs

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Outline

 Basics of probability theory  DAGS, Markov(G), Bayesian networks  Graphoids: axioms of for inferring

conditional independence (CI)

 D-separation: Inferring CIs in graphs

(Darwiche chapter 4)

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Bayesian Networks: Representation

= P(S) P(C|S) P(B|S) P(X|C,S) P(D|C,B) lung Cancer Smoking X-ray Bronchitis Dyspnoea

P(D|C,B) P(B|S) P(S) P(X|C,S) P(C|S)

P(S, C, B, X, D)

Conditional Independencies Efficient Representation

Θ) (G, BN 

CPD:

C B D=0 D=1 0 0 0.1 0.9 0 1 0.7 0.3 1 0 0.8 0.2 1 1 0.9 0.1

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The causal interpretation

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Graphs convey set of independence statements

 Undirected graphs by graph separation  Directed graphs by graph’s d-separation  Goal: capture probabilistic conditional

independence by graph graphs.

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Use GeNie/Smile To create this network

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Outline

 Basics of probability theory  DAGS, Markov(G), Bayesian networks  Graphoids: axioms for inferring

conditional independence (CI)

 D-separation: Inferring CIs in graphs

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(Darwiche, chapter 4 Pearl, Chapter 3)

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R and C are independent given A

This independence follows from the Markov assumption

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Properties of Probabilistic independence

Symmetry:

I(X,Z,Y)  I(Y,Z,X)

Decomposition:

I(X,Z,YW) I(X,Z,Y) and I(X,Z,W)

Weak union:

I(X,Z,YW)I(X,ZW,Y)

Contraction:

I(X,Z,Y) and I(X,ZY,W)I(X,Z,YW)

Intersection:

I(X,ZY,W) and I(X,ZW,Y)  I(X,Z,YW)

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Pearl’s language: If two pieces of information are irrelevant to X then each one is irrelevant to X

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Example: Two coins (C1,C2,) and a bell (B)

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When there are no constraints

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Properties of Probabilistic independence

Symmetry:

I(X,Z,Y)  I(Y,Z,X)

Decomposition:

I(X,Z,YW) I(X,Z,Y) and I(X,Z,W)

Weak union:

I(X,Z,YW)I(X,ZW,Y)

Contraction:

I(X,Z,Y) and I(X,ZY,W)I(X,Z,YW)

Intersection:

I(X,ZY,W) and I(X,ZW,Y)  I(X,Z,YW)

Graphoid axioms: Symmetry, decomposition Weak union and contraction Positive graphoid: +intersection In Pearl: the 5 axioms are called Graphids, the 4, semi-graphois

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Outline

 Basics of probability theory  DAGS, Markov(G), Bayesian networks  Graphoids: axioms of for inferring

conditional independence (CI)

 D-separation: Inferring CIs in graphs

 I-maps, D-maps, perfect maps  Markov boundary and blanket  Markov networks

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What we know so far on BN?

 A probability distribution of a Bayesian network

having directed graph G, satisfies all the Markov assumptions of independencies.

 5 graphoid, (or positive) axioms allow inferring more

conditional independence relationship for the BN.

 D-separation in G will allows deducing easily many of

the inferred independencies.

 G with d-separation yields an I-MAP of the probability

distribution.

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d-speration

To test whether X and Y are d-separated by Z in dag G, we need to consider every path between a node in X and a node in Y, and then ensure that the path is blocked by Z.

A path is blocked by Z if at least one valve (node) on the path is ‘closed’ given Z.

A divergent valve or a sequential valve is closed if it is in Z

A convergent valve is closed if it is not on Z nor any of its descendants are in Z.

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No path Is active = Every path is blocked

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Bayesian Networks as i-maps

 E: Employment  V: Investment  H: Health  W: Wealth  C: Charitable

contributions

 P: Happiness

E E E C E V W C P H Are C and V d-separated give E and P? Are C and H d-separated?

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d-Seperation Using Ancestral Graph

 X is d-separated from Y given Z (<X,Z,Y>d) iff:

Take the ancestral graph that contains X,Y,Z and their ancestral subsets.

Moralized the obtained subgraph

Apply regular undirected graph separation

Check: (E,{},V),(E,P,H),(C,EW,P),(C,E,HP)?

E E E C E V W C P H

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Idsep(R,EC,B)?

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Idsep(C,S,B)=?

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Is S1 conditionally on S2 independent of S3 and S4 In the following Bayesian network?

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Outline

 Basics of probability theory  DAGS, Markov(G), Bayesian networks  Graphoids: axioms of for inferring conditional

independence (CI)

 D-separation: Inferring CIs in graphs

 Soundness, completeness of d-seperation  I-maps, D-maps, perfect maps  Construction a minimal I-map of a distribution  Markov boundary and blanket

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It is not a d-map

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Outline

 Basics of probability theory  DAGS, Markov(G), Bayesian networks  Graphoids: axioms of for inferring conditional

independence (CI)

 D-separation: Inferring CIs in graphs

 Soundness, completeness of d-seperation  I-maps, D-maps, perfect maps  Construction a minimal I-map of a distribution  Markov boundary and blanket

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Outline

 Basics of probability theory  DAGS, Markov(G), Bayesian networks  Graphoids: axioms of for inferring conditional

independence (CI)

 D-separation: Inferring CIs in graphs

 Soundness, completeness of d-seperation  I-maps, D-maps, perfect maps  Construction a minimal I-map of a distribution  Markov boundary and blanket

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So how can we construct an I-MAP of a probability distribution? And a minimal I-Map

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Perfect Maps for DAGs

 Theorem 10 [Geiger and Pearl 1988]: For any dag D

there exists a P such that D is a perfect map of P relative to d-separation.

 Corollary 7: d-separation identifies any implied

independency that follows logically from the set of independencies characterized by its dag.

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Outline

 Basics of probability theory  DAGS, Markov(G), Bayesian networks  Graphoids: axioms of for inferring conditional

independence (CI)

 D-separation: Inferring CIs in graphs

 Soundness, completeness of d-seperation  I-maps, D-maps, perfect maps  Construction a minimal I-map of a distribution  Markov boundary and blanket

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Blanket Examples What is a Markov blanket of C?

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Blanket Examples

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Markov Blanket

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Bayesian Networks as Knowledge-Bases

 Given any distribution, P, and an ordering we can

construct a minimal i-map.

 The conditional probabilities of x given its parents is

all we need.

 In practice we go in the opposite direction: the

parents must be identified by human expert… they can be viewed as direct causes, or direct influences.

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Pearl corollary 4

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Corollary 4: Given a dag G and a probability distribution P, a necessary and sufficient Condition for G to be a Bayesian network of P is If all the Markovian assumptions are satisfied

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Markov Networks and Markov Random Fields (MRF)

Can we also capture conditional independence by undirected graphs? Yes: using simple graph separation

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Undirected Graphs as I-maps of Distributions

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Graphoids vs Undirected graphs

Symmetry: I(X,Z,Y)  I(Y,Z,X)

Decomposition: I(X,Z,YW) I(X,Z,Y) and I(X,Z,W)

Weak union: I(X,Z,YW)I(X,ZW,Y)

Contraction: I(X,Z,Y) and I(X,ZY,W)I(X,Z,YW)

Intersection: I(X,ZY,W) and I(X,ZW,Y)  I(X,Z,YW)

Symmetry: I(X,Z,Y)  I(Y,Z,X) Decomposition: I(X,Z,YW) I(X,Z,Y) and I(X,Z,Y) Intersection: I(X,ZW,Y) and I(X,ZY,W)I(X,Z,YW)

Strong union: I(X,Z,Y)  I(X,ZW, Y) Transitivity: I(X,Z,Y)  exists t s.t. I(X,Z,t) or I(t,Z,Y)

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See Pearl’s book

Graphoids: Conditional Independence Seperation in Graphs

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Markov Networks

 An undirected graph G which is a minimal I-map of

a probability distribution Pr, namely deleting any edge destroys its i-mappness relative to (undirected) seperation, is called a Markov network of P.

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The unusual edge (3,4) reflects the reasoning that if we fix the arrival time (5) the travel time (4) must depends on current time (3) slides2 COMPSCI 2020

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How can we construct a probability Distribution that will have all these independencies?

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So, How do we learn Markov networks From data?

Markov Random Field (MRF)

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Markov Networks

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Sample Applications for Graphical Models

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