Applied Bayesian Nonparametrics
3. Infinite Hidden Markov Models
Tutorial at CVPR 2012Erik Sudderth Brown 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 on underlying mode dynamics
Temporal Segmentation
Hidden Markov Model modes
observations
Outline Temporal Segmentation !!How many dynamical modes?
!!Mode persistence
!!Complex local dynamics
!!Multiple time series
Spatial Segmentation
!! Ising and Potts MRFs
!!Gaussian processes
Issue 1: How many modes?
•! Dirichlet process (DP): !! Mode space of unbounded size !! Model complexity adapts to
observations
•! Hierarchical: !! Ties mode transition
distributions !! Shared sparsity
Time
Mod
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
Sticky HDP-HMM
mode-specific base measure
Increased probability of self-transition
sticky original
Infinite HMM: Beal, et.al., NIPS 2002
Direct Assignment Sampler •! Marginalize:
!! Transition densities !! Emission parameters
•! Sequentially sample:
Conjugate base measure "" closed form
Chinese restaurant
prior likelihood
Collapsed Gibbs Sampler
Splits true mode, hard to
merge
•! Approximate HDP: "! Average transition density "! (" transition densities)
•! Sample:
Blocked Resampling HDP-HMM weak limit approximation HDP-HMM weak limit approximation
•! Compute backwards messages:
•! Block sample as:
" Average transition density " (" transition densities)
Hyperparameters •! Place priors on hyperparameters and infer them from data •! Weakly informative priors •! All results use the same settings
hyperparameters
can be set using the data
Related self-transition parameter: Beal, et.al., NIPS 2002
HDP-HMM: Multimodal Emissions
•! Approximate multimodal emissions with DP mixture
•! Temporal mode persistence disambiguates model
modes
mixture components
observations
John Jane Bob John Bob
Ji l 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
Overall DER
Best DER
Worst DER
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 DERsIC
SI D
ERs
! "! #! $! %! &!!
"!
#!
$!
%!
&!
'()*+,-./01
234
'()*+,-./0
1
= set of dynamic parameters
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:
State space models
VAR processes
•! Vector autoregressive (VAR) process:
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%
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 determine
which dynamical modes are used
•! Beta process prior: !! Encourages sharing !! Unbounded features
Features/Modes
Sequ
ence
s
!
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