Graphical Models - Department of Systems Engineering and...

Graphical ModelsA Brief Introduction

Reference: Pattern Recognition and Machine Learningby C.M. Bishop, SpringerChapter 8.2

https://www.microsoft.com/en‐us/research/wp‐content/uploads/2016/05/Bishop‐PRML‐sample.pdf

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ProbabilisticModel

Real WorldData

P(Data | Parameters)

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ProbabilisticModel

Real WorldData


P(Parameters | Data)

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ProbabilisticModel

Real WorldData



Generative Model, Probability

Inference, Statistics

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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 Hang Seng 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

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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

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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

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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

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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?

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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

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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

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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

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Examples of 3‐way Graphical Models

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

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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

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A B

C

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

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

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Directed Graphical Models

A B

C

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

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A B

C

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


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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 ) )


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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

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ProbabilisticModel

Real WorldData



Generative Model, Probability

Inference, Statistics

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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 ()

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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

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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

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Binomial Likelihood Examples

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Graphical Models

• Left – data points are conditionally independent given

• Right – plate notation (same model as left)repeated nodes are inside a box (plate)number in lower right hand corner , specifies the number of repetitions of the node

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• Represent using a graphical model:

• Assume each data case was generated independently but from the same distribution

• Data cases are only independent conditional on the parameters

• Marginally, the data cases are dependent• The order in which the data cases arrive makes

no difference to the benefits about (all orderings have same sufficient statistics) data is exchangeable

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Graphical Models

• Avoid visual clutter:use a form of syntactic sugar, called plates

• Draw a little box around the repeated variables• With the convention that nodes within the box is

repeated when the model is unrolled• Bottom right corner of the box: number of copies

or repetitions• The corresponding joint distribution has the form:

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Plate Notation

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:

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Graphical Model for Multinomial

w1

= [ p(w1), p(w2),….. p(wk) ]

w2 wn

Parameters

Observed data

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“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

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Learning in Graphical Models

wi

i=1:n

Data = D = {w1,…wn}

Model parameters

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

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Maximum Likelihood (ML) Principle

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

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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( )

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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

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Graphical Models - Department of Systems Engineering and...

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