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Hidden Markov Model -- Probabilistic Graphical Model Perspective Rui Li
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 )|(
