Applied Bayesian Nonparametrics 3. Infinite Hidden Markov Models - - PowerPoint PPT Presentation

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Applied Bayesian Nonparametrics 3. Infinite Hidden Markov Models - - PowerPoint PPT Presentation

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Applied Bayesian Nonparametrics 3. Infinite Hidden Markov Models Tutorial at CVPR 2012 Erik Sudderth Brown University Work by E. Fox, E. Sudderth, M. Jordan, & A. Willsky AOAS 2011: A Sticky HDP-HMM with Application to Speaker Diarization

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SLIDE 1

Applied Bayesian Nonparametrics

  • 3. Infinite Hidden Markov Models

Tutorial at CVPR 2012 Erik 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

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SLIDE 2

Observations True mode sequence

  • ! Markov switching

models for time series data

  • ! Cluster based on

underlying mode dynamics

Temporal Segmentation

Hidden Markov Model

modes

  • bservations
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SLIDE 3

Outline

Temporal Segmentation !! How many dynamical modes? !! Mode persistence !! Complex local dynamics !! Multiple time series Spatial Segmentation !! Ising and Potts MRFs !! Gaussian processes

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SLIDE 4

Hidden Markov Models

Time Mode

modes

  • bservations
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SLIDE 5

Hidden Markov Models

Time

modes

  • bservations
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SLIDE 6

Hidden Markov Models

Time

modes

  • bservations
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SLIDE 7

Hidden Markov Models

Time

modes

  • bservations
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SLIDE 8

Issue 1: How many modes?

  • ! Dirichlet process (DP):

!! Mode space of unbounded size !! Model complexity adapts to

  • bservations
  • ! Hierarchical:

!! Ties mode transition distributions !! Shared sparsity

Time Mode

Infinite HMM: Beal, et.al., NIPS 2002 HDP-HMM: Teh, et. al., JASA 2006

Hierarchical Dirichlet Process HMM

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SLIDE 9
  • ! Global transition distribution:!

HDP-HMM

sparsity of ! is shared

  • ! Mode-specific transition distributions:!

Hierarchical Dirichlet Process HMM

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SLIDE 10

Issue 2: Temporal Persistence

Hidden Markov Model

True mode sequence HDP-HMM inferred mode sequence

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Sticky HDP-HMM

Time Mode

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SLIDE 12

Sticky HDP-HMM

mode-specific base measure Increased probability of self-transition sticky

  • riginal

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

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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

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SLIDE 14
  • ! 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:
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SLIDE 15

Results: Gaussian Emissions

Blocked sampler HDP-HMM Sticky HDP-HMM Sequential sampler

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SLIDE 16

Sticky HDP-HMM HDP-HMM

Results: Fast Switching

Observations True mode sequence

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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

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SLIDE 18

HDP-HMM: Multimodal Emissions

  • ! Approximate multimodal

emissions with DP mixture

  • ! Temporal mode persistence

disambiguates model

modes mixture components

  • bservations
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SLIDE 19

John Jane Bob John B

  • b

J i l l

1 2 3 4 5 6 7 8 9 10 x 104
  • 30
  • 20
  • 10
10 20 30 40

Speaker Diarization

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SLIDE 20

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%

10 20 30 40 50 10 20 30 40 50

Sticky DERs ICSI DERs

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

'()*+,-./01 234'()*+,-./01

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Results: Meeting 1

Sticky DER = 1.26% ICSI DER = 7.56%

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Results: Meeting 18

Sticky DER = 20.48% ICSI DER = 22.00% 4.81%

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= 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

  • bservation dynamics

Switching Dynamical Processes

modes

  • bservations
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Linear Dynamical Systems

  • ! State space LTI model:
  • ! Vector autoregressive (VAR) process:
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SLIDE 25

Linear Dynamical Systems

  • ! State space LTI model:

State space models VAR processes

  • ! Vector autoregressive (VAR) process:
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Switching Dynamical Systems

Switching linear dynamical system (SLDS): Switching VAR process:

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HDP-AR-HMM and HDP-SLDS

HDP-AR-HMM HDP-SLDS

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SLIDE 28

Dancing Honey Bees

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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%

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SLIDE 30

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

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SLIDE 31

IBP-AR-HMM

  • !

Latent features determine which dynamical modes are used

  • !

Beta process prior: !! Encourages sharing !! Unbounded features

Features/Modes Sequences

!

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Motion Capture

CMU MoCap: http://mocap.cs.cmu.edu/

6 videos of exercise routines:

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Library of MoCap Behaviors

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