Post on 05-Apr-2018
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
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 2/40
Observations
True mode sequence•! Markov switching
models for time
series data
•! Cluster based onunderlying mode
dynamics
Temporal Segmentation
Hidden Markov Modelmodes
observations
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 3/40
Outline
Temporal Segmentation!! How many dynamical modes?
!! Mode persistence
!! Complex local dynamics
!! Multiple time series
Spatial Segmentation
!! Ising and Potts MRFs
!! Gaussian processes
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 4/40
Hidden Markov Models
Time
M o d e
modes
observations
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 5/40
Hidden Markov Models
Time
modes
observations
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 6/40
Hidden Markov Models
Time
modes
observations
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 7/40
Hidden Markov Models
Time
modes
observations
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 8/40
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
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 9/40
•!Global transition distribution:
HDP-HMM
sparsity of! is shared
•!Mode-specific transition distributions:
Hierarchical Dirichlet Process HMM
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 10/40
Issue 2: Temporal Persistence
Hidden Markov Model
True mode sequenceHDP-HMM inferred mode sequence
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 11/40
Sticky HDP-HMM
Time
M o d e
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 12/40
Sticky HDP-HMM
mode-specific base measure
Increased probability of self-transition
stickyoriginal
Infinite HMM: Beal, et.al., NIPS 2002
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 13/40
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
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 14/40
•!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:
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 15/40
Results: Gaussian Emissions
Blockedsampler
HDP-HMM Sticky HDP-HMM
Sequential
sampler
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 16/40
StickyHDP-HMM
HDP-HMM
Results: Fast Switching
Observations
True modesequence
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 17/40
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
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 18/40
HDP-HMM: Multimodal Emissions
•!Approximate multimodalemissions with DP mixture
•!Temporal mode persistencedisambiguates model
modes
mixturecomponents
observations
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 19/40
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
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 20/40
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
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 21/40
Results: Meeting 1
Sticky DER = 1.26%
ICSI DER = 7.56%
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 22/40
Results: Meeting 18
Sticky DER = 20.48%
ICSI DER = 22.00%
4.81%
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 23/40
= 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
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 24/40
Linear Dynamical Systems
•! State space LTI model:
•!Vector autoregressive (VAR) process:
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 25/40
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 26/40
Switching Dynamical Systems
Switching linear dynamicalsystem (SLDS):
Switching VAR process:
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 27/40
HDP-AR-HMM and HDP-SLDS
HDP-AR-HMM HDP-SLDS
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 28/40
Dancing Honey Bees
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 29/40
Honey Bee Results: HDP-AR(1)-HMM
Sequence 1 Sequence 2 Sequence 3
HDP-AR-HMM: 88.1%SLDS [Oh]: 93.4%
HDP-AR-HMM: 92.5%SLDS [Oh]: 90.2%
HDP-AR-HMM: 88.2%SLDS [Oh]: 90.4%
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 30/40
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
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 31/40
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
!
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 32/40
Motion Capture
CMU MoCap: http://mocap.cs.cmu.edu/6 videos of exercise routines:
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 33/40
Library of MoCap Behaviors
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 34/40
!"#$%&'()*+$,-'-.*'$ /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#
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 35/40
!"#$%&'()'!)%&*++,#-.+/)
,&.(19*2*:$%&'()*+$;*)-9&12.$
I(2+*%(:)#(/#)','2*#9'3+$%(0)#+20())#+,,#$%&'()#
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 36/40
!"#$%&'()'!)%&*++,#-.+/)
<&=)'$>?&'()$
<0(*(2(,#0'JA%0')#)K%*23%:5#,%53*#(:L(//#+*#)*+0*#+:&#M%:%)3#
,&.(19*2*:$%&'()*+$;*)-9&12.$
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 37/40
!"#$%&'()'!)%&*++,#-.+/)
@A*+$B2&:=*$
<%NN+#:''&)#23'')'-#C+:&K%23#:''&)#O',,6#*(#9'5%:#40'4+0+*%(:#
,&.(19*2*:$%&'()*+$;*)-9&12.$
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 38/40
!"#$%&'()'!)%&*++,#-.+/)
>*'$@9*+$
;(*3#<%NN+)#+:&#;0(K:%')#:''&#*(#9'#9+G'&#*(#2(:2,A&'#40'4+0+*%(:#
,&.(19*2*:$%&'()*+$;*)-9&12.$
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 39/40
7/31/2019 BNP3hdphmm Cvpr2012 Applied Bayesian Nonparametrics
http://slidepdf.com/reader/full/bnp3hdphmm-cvpr2012-applied-bayesian-nonparametrics 40/40