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

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

Temporal Segmentation • ! Markov switching True mode sequence Observations models for time series data • ! Cluster based on underlying mode dynamics modes Hidden Markov Model 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 modes Time observations Mode

Hidden Markov Models modes Time observations

Hidden Markov Models modes Time observations

Hidden Markov Models modes Time observations

Issue 1: How many modes? Infinite HMM: Beal, et.al., NIPS 2002 HDP-HMM: Teh, et. al., JASA 2006 Time Hierarchical Dirichlet Process HMM Mode • ! Dirichlet process (DP): ! ! Mode space of unbounded size ! ! Model complexity adapts to observations • ! Hierarchical: ! ! Ties mode transition distributions ! ! Shared sparsity

HDP-HMM Hierarchical Dirichlet Process HMM • ! Global transition distribution: ! • ! Mode-specific transition distributions: ! sparsity of ! is shared

Issue 2: Temporal Persistence HDP-HMM inferred mode sequence True mode sequence Hidden Markov Model

� Sticky � � HDP-HMM Time Mode

� Sticky � � HDP-HMM sticky original mode-specific base measure Increased probability of self-transition Infinite HMM: Beal, et.al., NIPS 2002

Direct Assignment Sampler • ! Marginalize: Collapsed Gibbs Sampler ! ! Transition densities ! ! Emission parameters Chinese restaurant • ! Sequentially sample: prior likelihood Conjugate base measure " " closed form Splits true mode, hard to merge

Blocked Resampling HDP-HMM weak limit approximation • ! Compute backwards messages: • ! Approximate HDP: HDP-HMM weak limit approximation " ! Average transition density " ! ( " transition densities) • ! Sample: • ! Block sample as:

Results: Gaussian Emissions HDP-HMM Sticky HDP-HMM Blocked sampler Sequential sampler

Results: Fast Switching Sticky Observations HDP-HMM True mode HDP-HMM sequence

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 modes mixture components observations • ! Approximate multimodal emissions with DP mixture • ! Temporal mode persistence disambiguates model

Speaker Diarization 40 30 20 10 0 -10 -20 -30 0 1 2 3 4 5 6 7 8 9 10 x 10 4 J B i o Bob John Jane John l b l

Results: 21 meetings 50 &! 234 � '()*+,-./01 40 %! ICSI DERs 30 $! 20 #! 10 "! 0 ! 0 10 20 30 40 50 ! "! #! $! %! &! Sticky DERs '()*+,-./01 Overall Best Worst DER DER DER Sticky HDP-HMM 17.84% 1.26% 34.29% Non-Sticky HDP- 23.91% 6.26% 46.95% HMM ICSI 18.37% 4.39% 32.23%

Results: Meeting 1 Sticky DER = 1.26% ICSI DER = 7.56%

Results: Meeting 18 4.81% Sticky DER = 20.48% ICSI DER = 22.00%

Issue 3: Complex Local Dynamics = set of dynamic • ! Discrete clusters may parameters not accurately capture high-dimensional data • ! Autoregressive HMM: Discrete-mode switching of smooth observation dynamics modes Switching Dynamical Processes observations

Linear Dynamical Systems • ! State space LTI model: • ! Vector autoregressive (VAR) process:

Linear Dynamical Systems • ! State space LTI model: State space models VAR processes • ! Vector autoregressive (VAR) process:

Switching Dynamical Systems Switching linear dynamical system (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% HDP-AR-HMM: 92.5% HDP-AR-HMM: 88.2% SLDS [Oh]: 93.4% SLDS [Oh]: 90.2% 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 Features/Modes Sequences ! • ! Beta process prior: ! ! Encourages sharing ! ! Unbounded features

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

Library of MoCap Behaviors

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