Hidden Markov Model -- Probabilistic Graphical Model Perspective

Rui Li

Resources

• Textbook and Tutorial

Resources

• Software

– Hidden Markov Model (HMM) Matlab Toolbox

• By Kevin Murphy

– GraphLab

• By CMU

– Hidden Markov Model Toolkit (HTK)

• C Libraries

Dynamic Phenomena

• Speech Recognition

Dynamic Phenomena

• Body Motion Tracking

Dynamic Phenomena

• Stock Prediction

Dynamic Phenomena

• Climate Change

Bioinformatics

• DNA Sequences

Outline

• Lecture One

– HMMs as Probabilistic Graphical Models

• Motivation

• Algebraic representation

• Graphical representation

• Lecture Two

– HMMs with Inference and Learning

• Message Passing (Forward-Backward)

• Expectation-Maximization (Baum-Welch)

• Application Demos

Motivation

• A simple graphical model

)|( XYP

Observed Unknown

Motivation

• An Example

x y

?

Motivation

• Inference

)(

)|()(

)(

),()|(

YP

XYPXP

YP

YXPYXP

Posterior probability

Prior probability Noise model

Constant

Curse of Dimensionality

80000100100 2256|)(| XP

x

Size of the lookup table of )(XP

Probabilistic Graphical Model

• The basic idea

– has some locality properties encoded by graphs

)(XP

Pixel 1 Pixel2

Pixel 3

Object Tracking

},,...,,{ 121 TT xxxxX

},,...,,{ 121 TT yyyyY tx location at time t

ty sensor measurement at time t

1000T

1000)1010(|)|(| YXP

Computation Complexity:

)|( YXPInference:

Probabilistic Graphical Model

),( YXP

Y

)|( YXP

)},{( tt yx

)|( YXP

• PDF Representation

–

• Inference

– Given

– Use to solve problems

• Learning

– Given

– Fit

Hidden Markov Models

• Definition

– are a HMM, if

• is a Markov process

• only depends on

Ttt yx ...1},{

X

tytx

)|(),,...,,,,...,,|( 11121 ttTTtttt xyPxxxxxxxyP

)|(),,...,,...,,|( 111121 ttTTttt xxPxxxxxxxP

Hidden Markov Models

• Representation

– Claim: for as a HMM

Ttt yx ...1},{

T

t

tttt xyPxxPYXP1

1 )|()|(),(

)|()(),( XYPXPYXP

T

t

tt

TTTT

TTTTTT

T

xxP

xPxxPxxPxxP

xPxxPxxxxPxxxxP

xxxPXP

1

1

112211

1121321121

21

)|(

)()|()...|()|(

)()|()...,...,,|(),...,|(

),...,,()(

T

t

tt

TTTT

TTTTTT

T

xyP

xyPxyPxyPxyP

XyPXyyPXyyyyPXyyyyP

XyyyPXYP

1

112211

1121321121

21

)|(

)|()|()...|()|(

)|(),|()...,,...,,|(),,...,|(

)|,...,()|(

Proof:

Hidden Markov Models

• Representation

– Claim: for as a HMM

Ttt yx ...1},{

T

t

tttt xyPxxPYXP1

1 )|()|(),(

Computational Complexity:

10001000 100100|),(| YXPbefore claim:

after claim: 21002000|),(| YXP

Hidden Markov Models

• Representation

– Claim: for as a HMM

Ttt yx ...1},{

T

t

tttt xyPxxPYXP1

1 )|()|(),(

Statistical queries:

)|(),|( 121 ttttt xxPxxxP

),|( 32 ttt xxxP

)|(

)|,(

)|()|(

),|(),,|(

),|,(

2

21

211

321321

321

1

1

1

1

tt

x

ttt

tt

x

tt

ttt

x

tttt

x

tttt

xxP

xxxP

xxPxxP

xxxPxxxxP

xxxxP

t

t

t

t

HMMs and Graphical Models

• Definition

– The graph represents a HMM is

• a chain of

• connects to

tx

tytx

HMMs and Graphical Models

• Theorem (Hammersley-Clifford)

– Given any random variables , and

iff separates and in the graph

A B C

)|(),|( BCPBACP

B A C

)|(),|( 26216 xxPxxxP

),|( 546 yyyP

HMMs and Graphical Models

• Graph & Factorization

T

t

tttt xyPxxPYXP1

1 )|()|(),(

HMMs and Graphical Models

Algebraic decomposition

Independence relationship

Graph

Probabilistic Graphical Model

),( YXP

Y

)|( YXP

)},{( tt yx

)|( YXP

• PDF Representation

–

• Inference

– Given

– Use to solve problems

• Learning

– Given

– Fit

Inference

• MAP (Maximum A Posteriori)

• Marginalization

Henceforth, “inference”== marginalization

)|(maxarg* YXPXX

tYxP t )|(