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Introduction to Machine Learning Probabilistic graphical models Yifeng Tao School of Computer Science Carnegie Mellon University Slides adapted from Eric Xing, Matt Gormley Yifeng Tao Carnegie Mellon University 1 Recap of Basic Probability

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Probabilistic graphical models

Yifeng Tao School of Computer Science Carnegie Mellon University Slides adapted from Eric Xing, Matt Gormley

Carnegie Mellon University 1 Yifeng Tao

Introduction to Machine Learning

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SLIDE 2

Recap of Basic Probability Concepts

  • Representation: the joint probability distribution on multiple binary

variables?

  • State configurations in total: 28
  • Are they all needed to be represented?
  • Do we get any scientific/medical insight?
  • Learning: where do we get all this probabilities?
  • Maximal-likelihood estimation?
  • Inference: If not all variables are observable, how to compute the

conditional distribution of latent variables given evidence?

  • Computing p(H|A) would require summing over all 26 configurations of the

unobserved variables

Yifeng Tao Carnegie Mellon University 2

[Slide from Eric Xing.]

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SLIDE 3

Graphical Model: Structure Simplifies Representation

  • Dependencies among variables

Yifeng Tao Carnegie Mellon University 3

[Slide from Eric Xing.]

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SLIDE 4

Probabilistic Graphical Models

  • If Xi’s are conditionally independent (as described by a PGM), the

joint can be factored to a product of simpler terms, e.g.,

  • Why we may favor a PGM?
  • Incorporation of domain knowledge and causal (logical) structures
  • 2+2+4+4+4+8+4+8=36, an 8-fold reduction from 28 in representation cost!

Yifeng Tao Carnegie Mellon University 4

[Slide from Eric Xing.]

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SLIDE 5

Two types of GMs

  • Directed edges give causality relationships (Bayesian Network or

Directed Graphical Model):

  • Undirected edges simply give correlations between variables

(Markov Random Field or Undirected Graphical model):

Yifeng Tao Carnegie Mellon University 5

[Slide from Eric Xing.]

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SLIDE 6

Bayesian Network

  • Definition:
  • It consists of a graph G and the conditional probabilities P
  • These two parts full specify the distribution:
  • Qualitative Specification: G
  • Quantitative Specification: P

Yifeng Tao Carnegie Mellon University 6

[Slide from Eric Xing.]

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SLIDE 7

Where does the qualitative specification come from?

  • Prior knowledge of causal relationships
  • Learning from data (i.e. structure learning)
  • We simply prefer a certain architecture (e.g. a layered graph)
  • …

Yifeng Tao Carnegie Mellon University 7

[Slide from Matt Gormley.]

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SLIDE 8

Quantitative Specification

  • Example: Conditional probability tables (CPTs) for discrete random

variables

Yifeng Tao Carnegie Mellon University 8

[Slide from Eric Xing.]

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Quantitative Specification

  • Example: Conditional probability density functions (CPDs) for

continuous random variables

Yifeng Tao Carnegie Mellon University 9

[Slide from Eric Xing.]

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Observed Variables

  • In a graphical model, shaded nodes are “observed”, i.e. their

values are given

Yifeng Tao Carnegie Mellon University 10

[Slide from Matt Gormley.]

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GMs are your old friends

  • Density estimation
  • Parametric and nonparametric methods
  • Regression
  • Linear, conditional mixture, nonparametric
  • Classification
  • Generative and discriminative approach
  • Clustering

Yifeng Tao Carnegie Mellon University 11

[Slide from Eric Xing.]

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SLIDE 12

What Independencies does a Bayes Net Model?

  • Independency of X and Z given Y?

P(X|Y)P(Z|Y) = P(X,Z|Y)

  • Three cases of interest...
  • Proof?

Yifeng Tao Carnegie Mellon University 12

[Slide from Matt Gormley.]

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SLIDE 13

The “Burglar Alarm” example

  • Your house has a twitchy burglar alarm that is also sometimes

triggered by earthquakes.

  • Earth arguably doesn’t care whether your house is currently being

burgled.

  • While you are on vacation, one of your neighbors calls and tells you

your home’s burglar alarm is ringing.

Yifeng Tao Carnegie Mellon University 13

[Slide from Matt Gormley.]

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Markov Blanket

  • Def: the co-parents of a node are the

parents of its children

  • Def: the Markov Blanket of a node is

the set containing the node’s parents, children, and co-parents.

  • Thm: a node is conditionally

independent of every other node in the graph given its Markov blanket

  • Example: The Markov Blanket of X6 is

{X3, X4, X5, X8, X9, X10}

Yifeng Tao Carnegie Mellon University 14

[Slide from Matt Gormley.]

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Markov Blanket

  • Example: The Markov Blanket of X6 is

{X3, X4, X5, X8, X9, X10}

Yifeng Tao Carnegie Mellon University 15

[Slide from Matt Gormley.]

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D-Separation

  • Thm: If variables X and Z are d-separated given a set of variables E

Then X and Z are conditionally independent given the set E

  • Definition:
  • Variables X and Z are d-separated given a set of evidence variables E

iff every path from X to Z is “blocked”.

Yifeng Tao Carnegie Mellon University 16

[Slide from Matt Gormley.]

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D-Separation

  • Variables X and Z are d-separated

given a set of evidence variables E iff every path from X to Z is “blocked”.

Yifeng Tao Carnegie Mellon University 17

[Slide from Eric Xing.]

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Machine Learning

Yifeng Tao Carnegie Mellon University 18

[Slide from Matt Gormley.]

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Recipe for Closed-form MLE

Yifeng Tao Carnegie Mellon University 19

[Slide from Matt Gormley.]

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Learning Fully Observed BNs

  • How do we learn these conditional and marginal distributions for a

Bayes Net?

Yifeng Tao Carnegie Mellon University 20

[Slide from Matt Gormley.]

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Learning Fully Observed BNs

  • Learning this fully observed Bayesian Network is equivalent to

learning five (small / simple) independent networks from the same data

Yifeng Tao Carnegie Mellon University 21

[Slide from Matt Gormley.]

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Learning Fully Observed BNs

Yifeng Tao Carnegie Mellon University 22

[Slide from Matt Gormley.]

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Learning Partially Observed BNs

  • Partially Observed Bayesian Network:
  • Maximal likelihood estimation à Incomplete log-likelihood
  • The log-likelihood contains unobserved latent variables
  • Solve with EM algorithm
  • Example: Gaussian Mixture Models (GMMs)

Yifeng Tao Carnegie Mellon University 23

[Slide from Eric Xing.]

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Inference of BNs

  • Suppose we already have the parameters of a Bayesian Network...

Yifeng Tao Carnegie Mellon University 24

[Slide from Matt Gormley.]

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Approaches to inference

  • Exact inference algorithms
  • The elimination algorithm à Message Passing
  • Belief propagation
  • The junction tree algorithms
  • Approximate inference techniques
  • Variational algorithms
  • Stochastic simulation / sampling methods
  • Markov chain Monte Carlo methods

Yifeng Tao Carnegie Mellon University 25

[Slide from Eric Xing.]

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Marginalization and Elimination

Yifeng Tao Carnegie Mellon University 26

[Slide from Eric Xing.]

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Marginalization and Elimination

Yifeng Tao Carnegie Mellon University 27

[Slide from Eric Xing.]

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SLIDE 28

Yifeng Tao Carnegie Mellon University 28

[Slide from Eric Xing.]

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  • Step 8: Wrap-up

Yifeng Tao Carnegie Mellon University 29

[Slide from Eric Xing.]

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Elimination algorithm

  • Elimination on trees is equivalent to message passing on branches
  • Message-passing is consistent in trees
  • Application: HMM

Yifeng Tao Carnegie Mellon University 30

[Slide from Eric Xing.]

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Gibbs Sampling

Yifeng Tao Carnegie Mellon University 31

[Slide from Matt Gormley.]

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Gibbs Sampling

Yifeng Tao Carnegie Mellon University 32

[Slide from Matt Gormley.]

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Gibbs Sampling

Yifeng Tao Carnegie Mellon University 33

[Slide from Matt Gormley.]

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Gibbs Sampling

  • Full conditionals only need to condition
  • n the Markov Blanket
  • Must be “easy” to sample from

conditionals

  • Many conditionals are log-concave and

are amenable to adaptive rejection sampling

Yifeng Tao Carnegie Mellon University 34

[Slide from Matt Gormley.]

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Take home message

  • Graphical models portrays the sparse dependencies of variables
  • Two types of graphical models: Bayesian network and Markov

random field

  • Conditional independence, Markov blanket, and d-separation
  • Learning fully observed and partially observed Bayesian networks
  • Exact inference and approximate inference of Bayesian networks

Carnegie Mellon University 35 Yifeng Tao

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References

  • Eric Xing, Ziv Bar-Joseph. 10701 Introduction to Machine Learning:

http://www.cs.cmu.edu/~epxing/Class/10701/

  • Matt Gormley. 10601 Introduction to Machine Learning:

http://www.cs.cmu.edu/~mgormley/courses/10601/index.html

Carnegie Mellon University 36 Yifeng Tao