Advanced Probabilistic Models for Generalized SSMs Speech and - PowerPoint PPT Presentation

Probabilistic Models State-Space Models Advanced Probabilistic Models for Generalized SSMs Speech and Language Nonlinear SSMs Bayesian Learning Mark Andrews Inferring Trajectories Gatsby Computational Neuroscience Unit Learning Example www.gatsby.ucl.ac.uk/ ~ mark Conclusion Monday 28 June, 2003 Title Page Abstract ◭ ◮ This project proposal applies to the statistical and machine learning models used in speech and language research. It aims to develop a more general theoretical Page 1 of 10 framework for the use of state-space models in speech and language, and to expand the set of models currently used in these fields. Go Back Full Screen Quit

1. Probabilistic Models Probabilistic Models State-Space Models Modeling speech and language demands the development and Generalized SSMs use of appropriate probabilistic models. Nonlinear SSMs These models are necessary for: Bayesian Learning 1. Machine Learning : Probabilistic models provide a gen- Inferring Trajectories eral means by which to derive and analyse algorithms. Learning Example 2. Neural-Network Modeling : Probabilistic models pro- Conclusion vide statistical interpretations of neural network models of language processing. Title Page ◭ ◮ 3. Data-analysis : Probabilistic models facilitate data-analysis in the experimental analysis of human language abilities. Page 2 of 10 Suitable models should describe, for example, the sequential Go Back and recursive structures found in the these domains. Full Screen Quit

2. State-Space Models Probabilistic Models State-Space Models In state-space models, the observed data are a function of a Generalized SSMs state-space that is evolving through time. Nonlinear SSMs Observed−Variables y t − 1 y t +1 y t Bayesian Learning Inferring Trajectories Learning Example Conclusion x t − 1 x t x t +1 Title Page Latent Variables ◭ ◮ Page 3 of 10 Examples of state-space models include Hidden Markov mod- Go Back els (HMMs), Kalman filter models (KFMs), their hybrids and their variants. Full Screen Despite their proven usefulness, many of these familiar models Quit have limitations.

3. Generalized State-Space Models Probabilistic Models State-Space Models To overcome limitations, powerful yet flexible generalizations Generalized SSMs of current models are needed. Nonlinear SSMs ❼ Powerful enough to model, for example, non-regular re- Bayesian Learning cursive structures, arbitrarily nonlinear functions, contin- Inferring Trajectories uous state topologies, and continuous time dynamics. Learning Example ❼ Flexible enough to yield tractable learning algorithms, Conclusion and allow for tractable inference. Title Page Possible candidates include nonlinear state-space models (NSSMs). ◭ ◮ Nonlinear state-space models can generalize both HMMs, KFMs and other state-space models. Page 4 of 10 Go Back Full Screen Quit

4. Nonlinear State-Space Models Probabilistic Models State-Space Models Generalized SSMs 1.5 1 0.8 P(y=2) Nonlinear SSMs 0.6 0.5 0.4 0.2 X 1 0 2 2 1 0 Bayesian Learning 0 −0.5 −1 −2 −2 X 1 X 0 −1 Inferring Trajectories −1.5 0.8 P(y=1) 0.6 0.4 −2 −1 0 1 0.2 X 0 Learning Example 2 2 20 1 0 0 −1 Conclusion 15 X 1 −2 −2 X 0 10 y 0 0.8 5 P(y=0) 0.6 0.4 0.2 0 2 2 Title Page 2 4 1 2 0 0 0 0 −2 −1 X 1 −2 −2 X 1 −2 −4 X 0 X 0 ◭ ◮ Figure 1: A nonlinear state-space model. The upper left figure shows the state-space Page 5 of 10 of a 2 -dimensional dynamical system, where arrows represent the rate and direction of the dynamics. The three figures on the right represent a multinomial output for Go Back the dynamical system. The three surfaces represent the probabilities for each of three discrete values. These surfaces sum to one at each point in the state-space. The lower Full Screen left figure represents a nonlinear scalar valued output-function. Quit

5. Bayesian Learning in Probabilistic Models Probabilistic Models State-Space Models Posterior : Generalized SSMs P( θ |D ) = P( θ ) � P( D , X | θ ) . Nonlinear SSMs P( D ) X Bayesian Learning Learning Methods : Inferring Trajectories Learning Example argmax P( θ |D ) MAP , θ Conclusion Q( θ ) log P( θ |D ) � argmin Variational Bayes , Q( θ ) Title Page Q θ N ◭ ◮ δ ( θ − ˜ � θ i ) ≈ P( θ |D ) MCMC . Page 6 of 10 i =1 Go Back Full Screen Quit

6. Inferring State-Space Trajectories Probabilistic Models State-Space Models Accurate inference of state-space trajectories is essential for Generalized SSMs learning. Nonlinear SSMs Bayesian Learning Inferring Trajectories 0.015 posterior−probability 0.01 Learning Example 0.005 Conclusion 0 2 Title Page 1.5 1 ◭ ◮ 0.5 10 8 0 6 4 state−space Page 7 of 10 2 −0.5 0 time Go Back Figure 2: Weighted particles for a one-dimensional state-space. The particles shown here represent P( x t | y 0: T , θ ) for 0 ≤ t ≤ 10 and from random parameter values for Full Screen a NSSM. The multi-modal and non-Gaussian distribution of the posterior is clearly evident. Quit

7. Example of MAP Learning in a NSSM Probabilistic Models State-Space Models 8 2 Generalized SSMs 7 1.5 6 1 Nonlinear SSMs X 1 X 1 5 0.5 4 0 Bayesian Learning −0.5 3 −1 2 −1 0 1 2 2 4 6 8 10 X 0 X 0 Inferring Trajectories 1 1 Symbol 1 Symbol 1 Learning Example Symbol 2 Symbol 2 0.8 0.8 Symbol 3 Symbol 3 0.6 0.6 Conclusion P P 0.4 0.4 0.2 0.2 0 0 −1 0 1 2 2 4 6 8 10 X 0 X 0 Title Page ◭ ◮ Figure 3: Example of learning a nonlinear state-space model for discrete output. The system that generated the data is shown on the left, while the learned system is on the Page 8 of 10 right. Notice how although the form of the dynamical mapping that is learned is almost identical to the data generator, the state-space itself has been rescaled considerably. Go Back The multinomial distributions also exhibit this rescaling. Full Screen Quit

8. Conclusion Probabilistic Models State-Space Models General and flexible models are necessary for all stages of lan- Generalized SSMs guage modelling. Nonlinear SSMs Nonlinear state-space models are a promising class of models. Bayesian Learning 1. Tractable learning and inference algorithms are feasible Inferring Trajectories for NSSMs. Learning Example 2. Recurrent neural network models can be given a proba- Conclusion bilistic interpretation in terms of NSSMs. Title Page 3. NSSMs provide methods for time-series analysis and psycho- ◭ ◮ linguistic data-analysis. Page 9 of 10 NSSMs have powerful modelling capabilities, yet have a prob- abilistic structure identical to other, more familiar, state-space Go Back models. Full Screen Quit

References Probabilistic Models State-Space Models [1] Lawrence R. Rabiner. A tutorial on Hidden Markov Models Generalized SSMs and selected applications in speech recognition. In Pro- Nonlinear SSMs ceedings of the IEEE , volume 77, pages 256–286, 1989. Bayesian Learning [2] Michael Isard and Andrew Blake. A smoothing filter for Inferring Trajectories Condensation. In Proceedings of the 5th European Con- Learning Example ference on Computer Vision , pages 767–781, 1998. Conclusion [3] Zoubin Ghahramani and Sam T. Roweis. Learning nonlin- ear dynamical systems using the EM algorithm. In Neural Title Page Information Processing Systems , vol. 11. 1998. ◭ ◮ [4] Harri Valpola and Juha Karhunen. An unsupervised en- Page 10 of 10 semble learning method for nonlinear dynamic state-space Go Back models. Neural Computation , 14(11), 2002. Full Screen Quit