Applied Bayesian Nonparametrics 3. Infinite Hidden Markov Models

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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.

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

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

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

4 Hidden Markov Models modes Time observations Mode

5 Hidden Markov Models modes Time observations

6 Hidden Markov Models modes Time observations

7 Hidden Markov Models modes Time observations

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

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

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

11 Sticky HDP-HMM Time Mode

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

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

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

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

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

17 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

18 HDP-HMM: Multimodal Emissions modes mixture components observations! Approximate multimodal emissions with DP mixture! Temporal mode persistence disambiguates model

19 Speaker Diarization x 10 4 Bob John J i l l B o b Jane John

20 Results: 21 meetings &! '()*+,-./01 %! $! #! "! ICSI DERs !! "! #! $! %! &! '()*+,-./ Sticky DERs 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%

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

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

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

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

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

26 Switching Dynamical Systems Switching linear dynamical system (SLDS): Switching VAR process:

27 HDP-AR-HMM and HDP-SLDS HDP-AR-HMM HDP-SLDS

28 Dancing Honey Bees

29 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%

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

31 IBP-AR-HMM! Latent features determine which dynamical modes are used Features/Modes Sequences!! Beta process prior:!! Encourages sharing!! Unbounded features

32 Motion Capture 6 videos of exercise routines: CMU MoCap:

33 Library of MoCap Behaviors

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Bayesian Nonparametric Learning of Complex Dynamical Phenomena

Duke University Department of Statistical Science Bayesian Nonparametric Learning of Complex Dynamical Phenomena Emily Fox Joint work with Erik Sudderth (Brown University), Michael Jordan (UC Berkeley),