Notes on Graphical Models

Notes on Graphical Models

Padhraic SmythDepartment of Computer Science

University of California, Irvine

ProbabilisticModel

Real WorldData

P(Data | Parameters)

ProbabilisticModel

Real WorldData


P(Parameters | Data)

ProbabilisticModel

Real WorldData



Generative Model, Probability

Inference, Statistics

Part 1: Review of Probability

Notation and Definitions

• X is a random variable– Lower-case x is some possible value for X– “X = x” is a logical proposition: that X takes value x– There is uncertainty about the value of X

• e.g., X is the Dow Jones index at 5pm tomorrow

• p(X = x) is the probability that proposition X=x is true– often shortened to p(x)

• If the set of possible x’s is finite, we have a probability distribution and p(x) = 1

• If the set of possible x’s is infinite, p(x) is a density function, and p(x) integrates to 1 over the range of X

Example

• Let X be the Dow Jones Index (DJI) at 5pm Monday August 22nd (tomorrow)

• X can take real values from 0 to some large number

• p(x) is a density representing our uncertainty about X– This density could be constructed from historical data, e.g.,

– After 5pm p(x) = 1 for some value of x (no uncertainty), once we hear from Wall Street what x is

Probability as Degree of Belief

• Different agents can have different p(x)’s– Your p(x) and the p(x) of a Wall Street expert might be

quite different– OR: if we were on vacation we might not have access to

stock market information• we would still be uncertain about p(x) after 5pm

• So we should really think of p(x) as p(x | BI)

– Where BI is background information available to agent I

– (will drop explicit conditioning on BI in notation)

• Thus, p(x) represents the degree of belief that agent I has in proposition x, conditioned on available background information

Comments on Degree of Belief

• Different agents can have different probability models– There is no necessarily “correct” p(x)– Why? Because p(x) is a model built on whatever assumptions or

background information we use– Naturally leads to the notion of updating

• p(x | BI) -> p(x | BI, CI)

• This is the subjective Bayesian interpretation of probability– Generalizes other interpretations (such as frequentist)– Can be used in cases where frequentist reasoning is not applicable– We will use “degree of belief” as our interpretation of p(x) in this

tutorial

• Note!– Degree of belief is just our semantic interpretation of p(x)– The mathematics of probability (e.g., Bayes rule) remain the same

regardless of our semantic interpretation

Multiple Variables

• p(x, y, z)– Probability that X=x AND Y=y AND Z =z– Possible values: cross-product of X Y Z

– e.g., X, Y, Z each take 10 possible values• x,y,z can take 103 possible values• p(x,y,z) is a 3-dimensional array/table

– Defines 103 probabilities• Note the exponential increase as we add more

variables

– e.g., X, Y, Z are all real-valued• x,y,z live in a 3-dimensional vector space• p(x,y,z) is a positive function defined over this space,

integrates to 1

Conditional Probability

• p(x | y, z)– Probability of x given that Y=y and Z = z– Could be

• hypothetical, e.g., “if Y=y and if Z = z”• observational, e.g., we observed values y and z

– can also have p(x, y | z), etc– “all probabilities are conditional probabilities”

• Computing conditional probabilities is the basis of many prediction and learning problems, e.g.,– p(DJI tomorrow | DJI index last week)– expected value of [DJI tomorrow | DJI index next week)– most likely value of parameter given observed data

Computing Conditional Probabilities

• Variables A, B, C, D– All distributions of interest related to A,B,C,D can be computed

from the full joint distribution p(a,b,c,d)

• Examples, using the Law of Total Probability

– p(a) = {b,c,d} p(a, b, c, d)

– p(c,d) = {a,b} p(a, b, c, d)

– p(a,c | d) = {b} p(a, b, c | d)

where p(a, b, c | d) = p(a,b,c,d)/p(d)

• These are standard probability manipulations: however, we will see how to use these to make inferences about parameters and unobserved variables, given data

Two Practical Problems

(Assume for simplicity each variable takes K values)

• Problem 1: Computational Complexity– Conditional probability computations scale as O(KN)

• where N is the number of variables being summed over

• Problem 2: Model Specification– To specify a joint distribution we need a table of O(KN) numbers

– Where do these numbers come from?

Two Key Ideas

• Problem 1: Computational Complexity– Idea: Graphical models

• Structured probability models lead to tractable inference

• Problem 2: Model Specification– Idea: Probabilistic learning

• General principles for learning from data

Part 2: Graphical Models

“…probability theory is more fundamentally concerned with the structure of reasoning and causation than with numbers.”

Glenn Shafer and Judea PearlIntroduction to Readings in Uncertain Reasoning,Morgan Kaufmann, 1990

Conditional Independence

• A is conditionally independent of B given C iff p(a | b, c) = p(a | c)

(also implies that B is conditionally independent of A given C)

• In words, B provides no information about A, if value of C is known

• Example:– a = “reading ability”– b = “height”– c = “age”

• Note that conditional independence does not imply marginal independence

Graphical Models

• Represent dependency structure with a directed graph– Node <-> random variable– Edges encode dependencies

• Absence of edge -> conditional independence– Directed and undirected versions

• Why is this useful?– A language for communication– A language for computation

• Origins: – Wright 1920’s– Independently developed by Spiegelhalter and Lauritzen in

statistics and Pearl in computer science in the late 1980’s

Examples of 3-way Graphical Models

A CB Marginal Independence:p(A,B,C) = p(A) p(B) p(C)


A

CB

Conditionally independent effects:p(A,B,C) = p(B|A)p(C|A)p(A)

B and C are conditionally independentGiven A

e.g., A is a disease, and we model B and C as conditionally independentsymptoms given A


A B

C

Independent Causes:p(A,B,C) = p(C|A,B)p(A)p(B)


A CB Markov dependence:p(A,B,C) = p(C|B) p(B|A)p(A)

Real-World Example

Monitoring Intensive-Care Patients• 37 variables• 509 parameters …instead of 237

(figure courtesy of Kevin Murphy/Nir Friedman)

PCWP CO

HRBP

HREKG HRSAT

ERRCAUTERHRHISTORY

CATECHOL

SAO2 EXPCO2

ARTCO2

VENTALV

VENTLUNG VENITUBE

DISCONNECT

MINVOLSET

VENTMACHKINKEDTUBEINTUBATIONPULMEMBOLUS

PAP SHUNT

ANAPHYLAXIS

MINOVL

PVSAT

FIO2PRESS

INSUFFANESTHTPR

LVFAILURE

ERRBLOWOUTPUTSTROEVOLUMELVEDVOLUME

HYPOVOLEMIA

CVP

BP

Directed Graphical Models

A B

C

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


A B

C

In general, p(X1, X2,....XN) = p(Xi | parents(Xi ) )



A B

C

• Probability model has simple factored form

• Directed edges => direct dependence

• Absence of an edge => conditional independence

• Also known as belief networks, Bayesian networks, causal networks

In general, p(X1, X2,....XN) = p(Xi | parents(Xi ) )


Graphical Models for Computation

PCWP CO

HRBP

HREKG HRSAT

ERRCAUTERHRHISTORY

CATECHOL

SAO2 EXPCO2

ARTCO2

VENTALV

VENTLUNG VENITUBE

DISCONNECT

MINVOLSET

VENTMACHKINKEDTUBEINTUBATIONPULMEMBOLUS

PAP SHUNT

ANAPHYLAXIS

MINOVL

PVSAT

FIO2PRESS

INSUFFANESTHTPR

LVFAILURE

ERRBLOWOUTPUTSTROEVOLUMELVEDVOLUME

HYPOVOLEMIA

CVP

BP

• Say we want to compute P(BP|Press)

• Law of total probability: -> must sum over all other variables -> exponential in # variables

• Factorization: -> joint distribution factors into smaller tables

• Can now sum over smaller tables, can reduce complexity dramatically

Example

D

A

B

C F

E

G

Example

D

A

B

C F

E

G

p(A, B, C, D, E, F, G) = p( variable | parents ) = p(A|B)p(C|B)p(B|D)p(F|E)p(G|E)p(E|D) p(D)

Example

D

A

B

c F

E

g

Say we want to compute p(a | c, g)

Example

D

A

B

c F

E

g

Direct calculation: p(a|c,g) = bdef p(a,b,d,e,f | c,g)

Complexity of the sum is O(K4)

Example

D

A

B

c F

E

g

Reordering (using factorization):

b p(a|b) d p(b|d,c) e p(d|e) f p(e,f |g)

Example

D

A

B

c F

E

g

Reordering:

bp(a|b) d p(b|d,c) e p(d|e) p(e|g)

p(d|g)

Example

D

A

B

c F

E

g

Reordering:

bp(a|b) d p(b|d,c) p(d|g)

p(b|c,g)

Example

D

A

B

c F

E

g

Reordering:

bp(a|b) p(b|c,g)

p(a|c,g) Complexity is O(K), compared to O(K4)

Graphs with “loops”

D

A

B

C F

E

G

Message passing algorithm does not work whenthere are multiple paths between 2 nodes

Graphs with “loops”

D

A

B

C F

E

G

General approach: “cluster” variablestogether to convert graph to a tree

Reduce to a Tree

D

A

B, E

C F G

Probability Calculations on Graphs

• General algorithms exist - beyond trees – Complexity is typically O(m (number of parents ) )

(where m = arity of each node)– If single parents (e.g., tree), -> O(m)– The sparser the graph the lower the complexity

• Technique can be “automated”– i.e., a fully general algorithm for arbitrary graphs– For continuous variables:

• replace sum with integral– For identification of most likely values

• Replace sum with max operator

Part 3: Learning with Graphical Models

Further Reading:

M. Jordan, Graphical models, Statistical Science: Special Issue on Bayesian Statistics, vol. 19, no. 1, pp. 140-155, Feb. 2004

A.Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin,Bayesian Data Analysis (2nd ed), Chapman and Hall, 2004

ProbabilisticModel

Real WorldData



Generative Model, Probability

Inference, Statistics

The Likelihood Function

• Likelihood = p(data | parameters)

= p( D | )

= L ()

• Likelihood tells us how likely the observed data are conditioned on a particular setting of the parameters

• Details– Constants that do not involve can be dropped in defining L ()

– Often easier to work with log L ()

Comments on the Likelihood Function

• Constructing a likelihood function L () is the first step in

probabilistic modeling

• The likelihood function implicitly assumes an underlying probabilistic model M with parameters

• L () connects the model to the observed data

• Graphical models provide a useful language for constructing likelihoods

Binomial Likelihood• Binomial model

– N memoryless trials, 2 outcomes

– probability of success at each trial

• Observed data– r successes in n trials – Defines a likelihood:

L() = p(D | )

= p(successes) p(non-successes)

= r (1-) n-r

Binomial Likelihood Examples

Multinomial Likelihood• Multinomial model

– N memoryless trials, K outcomes

– Probability vector for outcomes at each trial

• Observed data– nj successes in n trials

– Defines a likelihood:

– Maximum likelihood estimates:

Graphical Model for Multinomial

w1

= [ p(w1), p(w2),….. p(wk) ]

w2 wn

Parameters

Observed data

“Plate” Notation

wi

i=1:n

Data = D = {w1,…wn}

Model parameters

Plate (rectangle) indicates replicated nodes in a graphical model

Variables within a plate are conditionally independent manner given parent

Learning in Graphical Models

wi

i=1:n

Data = D = {w1,…wn}

Model parameters

Can view learning in a graphical model as computing themost likely value of the parameter node given the data nodes

Maximum Likelihood (ML) Principle (R. Fisher ~ 1922)

wi

i=1:n

L () = p(Data | ) = p(yi | )

Maximum Likelihood: ML = arg max{ Likelihood() }

Select the parameters that make the observed data most likely

Data = {w1,…wn}

Model parameters

The Bayesian Approach to Learning

wi

i=1:n

Fully Bayesian: p( | Data) = p(Data | ) p() / p(Data)

Maximum A Posteriori: MAP = arg max{ Likelihood() x Prior() }

Prior() = p( )

Learning a Multinomial

• Likelihood: same as before

• Prior: p( ) = Dirichlet (1,…K)

proportional to

– Has mean Prior weight for j

Can set all j = for “uniform” prior

Dirichlet Shapes

From: http://en.wikipedia.org/wiki/Dirichlet_distribution

Bayesian Learning

• P( | D, ) is proportional to p(data | ) p() =

=

= Dirichlet(n1 + 1,…, nK +K )

Posterior mean estimate

Summary of Bayesian Learning

• Can use graphical models to describe relationships between parameters and data

• P(data | parameters) = Likelihood function

• P(parameters) = prior– In applications such as text mining, prior can be “uninformative”, i.e., flat– Prior can also be optimized for prediction (e.g., on validation data)

• We can compute P(parameters | data, prior) or a “point estimate” (e.g., posterior mode or mean)

• Computation of posterior estimates can be computationally intractable – Monte Carlo techniques often used

Notes on Graphical Models

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