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Slides Set 5: Probabilistic Networks Rina Dechter Darwiche - - PowerPoint PPT Presentation

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Algorithms for Reasoning with graphical models Slides Set 5: Probabilistic Networks Rina Dechter Darwiche chapter 3,4, Pearl: chapters 3 slides5 828X 2019 Outline Basics of probability theory DAGS, Markov(G), Bayesian networks

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Algorithms for Reasoning with graphical models

Slides Set 5: Probabilistic Networks

Rina Dechter

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Darwiche chapter 3,4, Pearl: chapters 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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Examples: Common Sense Reasoning

Zebra on Pajama: (7:30 pm): I told Susannah: you have a nice

pajama, but it was just a dress. Why jump to that conclusion?: 1. because time is night time. 2. certain designs look like pajama.

Cars going out of a parking lot: You enter a parking lot which is

quite full (UCI), you see a car coming : you think ah… now there is a space (vacated), OR… there is no space and this guy is looking and leaving to another parking lot. What other clues can we have?

Robot gets out at a wrong level: A robot goes down the elevator. stops at 2nd floor instead of ground floor. It steps out and should immediately recognize not being in the right level, and go back inside.

Turing quotes

If machines will not be allowed to be fallible they cannot be intelligent

(Mathematicians are wrong from time to time so a machine should also be allowed)

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Why/What/How Uncertainty?

 Why Uncertainty?

 Answer: It is abandant

 What formalism to use?

 Answer: Probability theory

 How to overcome exponential

representation?

 Answer: Graphs, graphs, graphs… to

capture irrelevance, independence

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Why Uncertainty?

AI goal: to have a declarative, model-based, framework that allows computer system to reason.

People reason with partial information

Sources of uncertainty:

Limitation in observing the world: e.g., a physician see symptoms and not exactly what goes in the body when he performs diagnosis. Observations are noisy (test results are inaccurate)

Limitation in modeling the world,

maybe the world is not deterministic.

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Alpha and beta are events

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Burglary is independent of Earthquake

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Earthquake is independent of burglary

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Example

P(B,E,A,J,M)=?

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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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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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The causal interpretation

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

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Example: Two coins and a bell

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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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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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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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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

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Blanket Examples

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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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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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Graphoids

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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Undirected Graphs as I-maps of Distributions

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Axiomatic Characterization of Graphs

Graph separation satisfies:

 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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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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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) slides5 828X 2019

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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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Examples of Bayesian and and Markov Networks

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Markov Networks

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Sample Applications for Graphical Models

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