Tutorial on Probabilistic Graphical Models: A Geometric ...Tutorial on Probabilistic Graphical...

Probabilistic Graphical ModelsReasoning Bayesian Networks by Hand

How to reason by a machine?

Tutorial on Probabilistic Graphical Models:A Geometric and Topological View

Qinfeng (Javen) Shi

7 July 2017

Qinfeng (Javen) Shi Tutorial on Probabilistic Graphical Models: A Geometric and Topological View

Schedule

Part 1: Representation and Basic (Javen)

Part 2: Inference (Chao)

Part 3: Advanced Inference (Chao)

Part 4: Learning (both parameters and structures) (Javen).

Part 1: Representation and Basics I

1 Probabilistic Graphical ModelsHistory and booksRepresentationsFactorisation and independences

2 Reasoning Bayesian Networks by HandFactorisation and ReasoningA universal way

3 How to reason by a machine?VE for marginal inference

History and booksRepresentationsFactorisation and independences

Scenario 1

Multiple problems (A,B, ...) affect each other

Joint optimal solution of all 6= the solutions of individuals

Scenario 2

Two variables X ,Y each taking 10 possible values.Listing P(X ,Y ) for each possible value of X ,Y requiresspecifying/computing 102 many probabilities.

What if we have 1000 variables each taking 10 possible values?⇒ 101000 many probabilities

⇒ Difficult to store, and query naively.

Scenario 2

What if we have 1000 variables each taking 10 possible values?

⇒ 101000 many probabilities

Scenario 2

What if we have 1000 variables each taking 10 possible values?⇒ 101000 many probabilities

Remedy

Structured Learning, specially Probabilistic Graphical Models(PGMs).

PGMs use graphs to represent the complex probabilisticrelationships between random variables.

P(A,B,C , ...)

Benefits:

compactly represent distributions of variables.

Relation between variables are intuitive (such as conditionalindependences)

have fast and general algorithms to query withoutenumeration. e.g. ask for P(A|B = b,C = c) or EP [f ]

An Example

Difficulty Intelligence

Letter

Intuitive

Example

Difficulty Intelligence

Letter

P(S | I)

P(J | L,S)

P(G | D,I)

P(H | G,J)

P(L | G)

Compact

History

Gibbs (1902) used undirected graphs in particles

Wright (1921,1934) used directed graph in genetics

In economists and social sci (Wold 1954, Blalock, Jr. 1971)

In statistics (Bartlett 1935, Vorobev 1962, Goodman 1970,Haberman 1974)

In AI, expert system (Bombal et al. 1972, Gorry and Barnett1968, Warner et al. 1961)

Widely accepted in late 1980s. Prob Reasoning in Intelli Sys(Pearl 1988), Pathfinder expert system (Heckerman et al.1992)

History

Hot since 2001. Flexible features and principled ways oflearning.CRFs (Lafferty et al. 2001), SVM struct (Tsochantaridis etal2004), M3Net (Taskar et al. 2004), DeepBeliefNet (Hinton etal. 2006)

Super-hot since 2010. Winners of a large number ofchallenges with big data.Google, Microsoft, Facebook all open new labs for it.

History

Good books

Chris Bishop’s book “Pattern Recognition and MachineLearning” (Graphical Models are in chapter 8, which isavailable from his webpage) ≈ 60 pages

Koller and Friedman’s “Probabilistic Graphical Models”> 1000 pages

Stephen Lauritzen’s “Graphical Models”

Michael Jordan’s unpublished book “An Introduction toProbabilistic Graphical Models”

· · ·

Representations

(a) Directed graph

(b) Undirected graph

(c) Factor graph

Nodes represent random variables

Edges reflect dependencies between variables

Factors explicitly show which variables are used in each factori.e. f1(A,B)f2(A,C )f3(B,C )

Example — Image Denoising

Denoising1

Applications in Vision and PR

Image denoising

Original CorrectedNoisy

Denoising

Real Applications

),( ii yxΦ

),( ji xxΨ

X ∗ = argmaxX P(X |Y )

1This example is from Tiberio Caetano’s short course: “Machine Learningusing Graphical Models”

Example —Human Interaction Recognition

Factorisation for Bayesian networks

Directed Acyclic Graph (DAG).Factorisation rule: P(x1, . . . , xn) =

∏ni=1 P(xi |Pa(xi ))

Pa(xi ) denotes parent of xi . e.g. (A,B) = Pa(C )

⇒ P(A,B,C ) = P(A)P(B|A)P(C |A,B)Acyclic: no cycle allowed. Replacing edge A→ C with C → A willform a cycle (loop i.e. A→ B → C → A), not allowed in DAG.

Factorisation for Markov Random Fields

Undirected Graph:Factorisation rule:P(x1, . . . , xn) = 1

∏c∈C ψc(Xc),

Z =∑

∏c∈C ψc(Xc),

where c is an index set of a clique (fully connected subgraph), Xc

is the set of variables indicated by c.

Consider Xc1 = {A,B},Xc2 = {A,C},Xc3 = {B,C}⇒ P(A,B,C ) = 1

Z ψc1(A,B)ψc2(A,C )ψc3(B,C )

Consider Xc = {A,B,C} ⇒ P(A,B,C ) = 1Z ψc(A,B,C ),

Factorisation for Markov Random Fields

Factor Graph:Factorisation rule:P(x1, . . . , xn) = 1

∏i fi (Xi ), Z =

∏i fi (Xi )

⇒ P(A,B,C ) = 1Z f1(A,B)f2(A,C )f3(B,C )

Inference and Learning

Given potentials and the graph, one can ask for:

x∗ = argmaxx P(x) MAP Inference

P(xc) =∑

xV/cP(x) Marginal Inference

How to get potentials and the graph? → Learning.

Independences

IndependenceA ⊥⊥ B ⇔ P(A,B) = P(A)P(B)

Conditional IndependenceA ⊥⊥ B|C ⇔ P(A,B|C ) = P(A|C )P(B|C )

From Graph to Independences

Case 1:

Question: B ⊥⊥ C?

Answer: No.

P(B,C ) =∑A

P(A,B,C )

P(B|A)P(C |A)P(A)

6= P(B)P(C ) in general

Case 1:

Question: B ⊥⊥ C?Answer: No.

P(B,C ) =∑A

P(A,B,C )

P(B|A)P(C |A)P(A)

6= P(B)P(C ) in general

Case 2:

Question: B ⊥⊥ C |A?

Answer: Yes.

P(B,C |A) =P(A,B,C )

=P(B|A)P(C |A)P(A)

= P(B|A)P(C |A)

Case 2:

Question: B ⊥⊥ C |A?Answer: Yes.

P(B,C |A) =P(A,B,C )

=P(B|A)P(C |A)P(A)

= P(B|A)P(C |A)

From graphs to independences

Case 3:

Question: B ⊥⊥ C , B ⊥⊥ C |A?

∵ P(A,B,C ) = P(B)P(C )P(A|B,C ),

∴ P(B,C ) =∑A

P(A,B,C )

P(B)P(C )P(A|B,C )

= P(B)P(C )

From graphs to independences

Case 3:

Question: B ⊥⊥ C , B ⊥⊥ C |A?

∵ P(A,B,C ) = P(B)P(C )P(A|B,C ),

∴ P(B,C ) =∑A

P(A,B,C )

P(B)P(C )P(A|B,C )

= P(B)P(C )

Factorisation and ReasoningA universal way

Bayesian networks

Example:

P(A,B,C ) = P(A)P(B|A)P(C |A,B)

Bayesian networks

Example:

P(A,B,C ) = P(A)P(B|A)P(C |A,B)

Reasoning with all variables

DAG tells us: P(A,B,C ) = P(A)P(B|A)P(C |A,B)

P(A = a|B = b,C = c) =?

All variables are involved in the query.

=P(A = a, B = b, C = c)

P(B = b, C = c)

=P(A = a)P(B = b|A = a)P(C = c|A = a, B = b)∑

A∈{¬a,a} P(A, B = b, C = c)

A∈{¬a,a} P(A)P(B = b|A)P(C = c|A, B = b)

=P(A = a)P(B = b|A = a)P(C = c|A = a, B = b)

P(A = ¬a)P(B = b|A = ¬a)P(C = c|A = ¬a, B = b) + P(A = a)P(B = b|A = a)P(C = c|A = a, B = b)

Reasoning with all variables

P(A = a|B = b,C = c) =?

All variables are involved in the query.

=P(A = a, B = b, C = c)

P(B = b, C = c)

A∈{¬a,a} P(A, B = b, C = c)

A∈{¬a,a} P(A)P(B = b|A)P(C = c|A, B = b)

=P(A = a)P(B = b|A = a)P(C = c|A = a, B = b)

P(A = ¬a)P(B = b|A = ¬a)P(C = c|A = ¬a, B = b) + P(A = a)P(B = b|A = a)P(C = c|A = a, B = b)

Reasoning with missing variable(s)

P(A = a|B = b) =?

C is missing in the query.

=P(A = a,B = b)

P(B = b)

∑C P(A = a,B = b,C )∑A,C P(A,B = b,C )

∑C P(A = a)P(B = b|A = a)P(C |A = a,B = b)∑

A,C P(A)P(B = b|A)P(C |A,B = b)

Reasoning with missing variable(s)

P(A = a|B = b) =?

C is missing in the query.

=P(A = a,B = b)

P(B = b)

∑C P(A = a,B = b,C )∑A,C P(A,B = b,C )

∑C P(A = a)P(B = b|A = a)P(C |A = a,B = b)∑

A,C P(A)P(B = b|A)P(C |A,B = b)

Example of 4WD

Someone finds that people who drive 4WDs vehicles (S) consumelarge amounts of gas (G ) and are involved in more accidents thanthe national average (A). They have constructed the Bayesiannetwork below (here t implies “true” and f implies “false”).

Figure: 4WD Bayesian network

Example of 4WD

P(¬g , a|s)? (i.e. P(G = ¬g ,A = a|S = s))

P(a|s)?

P(A|s)?

Example of 4WD

Someone else finds that there are two types of people that drive4WDs, people from the country (C ) and people with large families(F ). After collecting some statistics, here is the new Bayesiannetwork.

Example of 4WD

P(g ,¬a, s, c)?

P(g ,¬a|c , f )?

How to reason by hand?

A universal way: express the query probability in terms of the fulldistribution2, and then factorise it.

Step by step:

1 when you see a conditional distribution, break it into thenominator and the denominator.

2 when you see a distribution (may be from the nominatorand/or the denominator) with missing variable(s), rewrite it asa sum of the full distribution w.r.t. the missing variable(s).

3 After everything is expressed by the full distribution, factorisethe full distribution into local distributions (which are known).

2the joint distribution over all variables of your Bayesian networkQinfeng (Javen) Shi Tutorial on Probabilistic Graphical Models: A Geometric and Topological View

Step by step:

How to reason by a machine?VE for marginal inference

Issues?

Do you want to try this for 100 variables?

Hand-tiring for many variables, and it’s only for Bayesian Networks.

How to infer for other graphical models and how to do it in acomputer program?

Issues?

Do you want to try this for 100 variables?Hand-tiring for many variables, and it’s only for Bayesian Networks.

How to infer for other graphical models and how to do it in acomputer program?

Variable elimination: infer by eliminating variables (works for bothmarginal and MAP inference)

P(A) =∑S,G

P(A,S ,G )

=∑S,G

P(S)P(A|S)P(G |S)

P(S)P(A|S)(∑G

P(G |S)) =∑S

P(S)P(A|S)

VE for marginal inference

Step by step:

1 sum over missing variables (marginalisation) for the fulldistribution.

2 factorise the full distribution.

3 rearrange the sum operator to reduce the computation.

4 eliminate the variables.

That’s all

Thanks!

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