bnp3hdphmm cvpr2012 applied bayesian nonparametrics

7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics

http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 1/40

Applied Bayesian

Nonparametrics3. Infinite Hidden Markov Models

Tutorial at CVPR 2012

Erik SudderthBrown University

Work by E. Fox, E. Sudderth, M. Jordan, & A. Willsky

AOAS 2011: A Sticky HDP-HMM with Application to Speaker Diarization

IEEE TSP 2011 & NIPS 2008:Bayesian Nonparametric Inference of Switching

Dynamic Linear Models

NIPS 2009:Sharing Features among Dynamical Systems with Beta Processes



Observations

True mode sequence•! Markov switching

models for time

series data

•! Cluster based onunderlying mode

dynamics

Temporal Segmentation

Hidden Markov Modelmodes

observations



Outline

Temporal Segmentation!! How many dynamical modes?

!! Mode persistence

!! Complex local dynamics

!! Multiple time series

Spatial Segmentation

!! Ising and Potts MRFs

!! Gaussian processes



Hidden Markov Models

Time

M o d e

modes

observations




Time

modes

observations



Issue 1: How many modes?

•! Dirichlet process (DP):!! Mode space of unbounded size

!!Model complexity adapts toobservations

•! Hierarchical:!! Ties mode transition

distributions

!! Shared sparsity

Time

M o d e

Infinite HMM: Beal, et.al., NIPS 2002

HDP-HMM: Teh, et. al., JASA 2006

Hierarchical Dirichlet Process HMM



•!Global transition distribution:

HDP-HMM

sparsity of! is shared

•!Mode-specific transition distributions:

Hierarchical Dirichlet Process HMM



Issue 2: Temporal Persistence

Hidden Markov Model

True mode sequenceHDP-HMM inferred mode sequence



Sticky HDP-HMM

Time

M o d e



Sticky HDP-HMM

mode-specific base measure

Increased probability of self-transition

stickyoriginal

Infinite HMM: Beal, et.al., NIPS 2002



Direct Assignment Sampler

•! Marginalize:!! Transition densities

!! Emission parameters

•! Sequentially sample:

Conjugate basemeasure"

closed form

Chineserestaurant

prior likelihood

Collapsed Gibbs Sampler

Splits truemode, hard to

merge



•!Approximate HDP:"! Average transition density"! ("transition densities)

•!Sample:

Blocked Resampling

HDP-HMM weak limit approximationHDP-HMM weak limit approximation•!Compute backwards messages:

•!Block sample as:



Results: Gaussian Emissions

Blockedsampler

HDP-HMM Sticky HDP-HMM

Sequential

sampler



StickyHDP-HMM

HDP-HMM

Results: Fast Switching

Observations

True modesequence



Hyperparameters

•! Place priors on hyperparameters and infer them from data•! Weakly informative priors

•! All results use the same settings

hyperparameters

can be setusing the data

Related self-transition parameter:Beal, et.al., NIPS 2002



HDP-HMM: Multimodal Emissions

•!Approximate multimodalemissions with DP mixture

•!Temporal mode persistencedisambiguates model

modes

mixturecomponents

observations



John JaneBob John

B

ob

J

il

l

0 1 2 3 4 5 6 7 8 9 10

x 104

-30

-20

-10

0

10

20

30

40

Speaker Diarization



Results: 21 meetings

OverallDER BestDER WorstDER

Sticky HDP-HMM 17.84% 1.26% 34.29%

Non-Sticky HDP-HMM

23.91% 6.26% 46.95%

ICSI 18.37% 4.39% 32.23%

0 10 20 30 40 500

10

20

30

40

50

Sticky DERs

I C S I D E R s

! "! #! $! %! &!!

"!

#!

$!

%!

&!

'()*+,-./01

2 3 4 ' ( ) * + , . / 0 1



Results: Meeting 1

Sticky DER = 1.26%

ICSI DER = 7.56%



Results: Meeting 18

Sticky DER = 20.48%

ICSI DER = 22.00%

4.81%



= set of dynamicparameters

Issue 3: Complex Local Dynamics

•! Discrete clusters may

not accurately capture

high-dimensional data

•! Autoregressive HMM:

Discrete-mode

switching of smooth

observation dynamics

Switching Dynamical

Processes

modes

observations



Linear Dynamical Systems

•! State space LTI model:

•!Vector autoregressive (VAR) process:



Switching Dynamical Systems

Switching linear dynamicalsystem (SLDS):

Switching VAR process:



HDP-AR-HMM and HDP-SLDS

HDP-AR-HMM HDP-SLDS



Dancing Honey Bees



Honey Bee Results: HDP-AR(1)-HMM

Sequence 1 Sequence 2 Sequence 3

HDP-AR-HMM: 88.1%SLDS [Oh]: 93.4%





Issue 4: Multiple Time Series

•! Goal:

!!Transfer knowledge between related time series

!! Allow each system to switch between an

arbitrarily large set of dynamical modes•! Method:

!!Beta process prior

!!Predictive distribution: Indian buffet process



IBP-AR-HMM

•! Latent features determinewhich dynamical modes

are used

•! Beta process prior:

!! Encourages sharing

!! Unbounded features

Features/Modes

S e q u e n c e s

!



Motion Capture

CMU MoCap: http://mocap.cs.cmu.edu/6 videos of exercise routines:



Library of MoCap Behaviors



!"#$%&'()*+$,-'-.*'$ /0122*$*'$-34$05$678$

!"#$%&'()#*(*+,-#."#/0(1#'+23#0'2%4'#2+*'5(067#8+*'5(06#,+9',)#:(*#40($%&'&#*(#;<=>??7#

@')A,*)#/0(1#>A53')#B#CA&&'0*3-#<D8E#F(0G)3(4-#8E<@#H".H7#



!"#$%&'()'!)%&*++,#-.+/)

,&.(19*2*:$%&'()*+$;*)-9&12.$

I(2+*%(:)#(/#)','2*#9'3+$%(0)#+20())#+,,#$%&'()#



!"#$%&'()'!)%&*++,#-.+/)

<&=)'$>?&'()$

<0(*(2(,#0'JA%0')#)K%*23%:5#,%53*#(:L(//#+*#)*+0*#+:&#M%:%)3#

,&.(19*2*:$%&'()*+$;*)-9&12.$



!"#$%&'()'!)%&*++,#-.+/)

@A*+$B2&:=*$

<%NN+#:''&)#23'')'-#C+:&K%23#:''&)#O',,6#*(#9'5%:#40'4+0+*%(:#

,&.(19*2*:$%&'()*+$;*)-9&12.$



!"#$%&'()'!)%&*++,#-.+/)

>*'$@9*+$

;(*3#<%NN+)#+:&#;0(K:%')#:''&#*(#9'#9+G'&#*(#2(:2,A&'#40'4+0+*%(:#

,&.(19*2*:$%&'()*+$;*)-9&12.$

bnp3hdphmm cvpr2012 applied bayesian nonparametrics

Documents