Inference in hidden Markov models /

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Bibliographic Details
Author / Creator:	Cappé, Olivier.
Imprint:	New York : Springer, c2005.
Description:	xvii, 652 p. : ill. ; 25 cm.
Language:	English
Series:	Springer series in statistics
Subject:	Markov processes. Markov processes.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/5750092

Hidden Bibliographic Details
Other authors / contributors:	Moulines, Eric. Rydén, Tobias, 1966-
ISBN:	9780387402642 0387402640

Table of Contents:

Preface
Contributors
1. Introduction
1.1. What Is a Hidden Markov Model?
1.2. Beyond Hidden Markov Models
1.3. Examples
1.3.1. Finite Hidden Markov Models
1.3.2. Normal Hidden Markov Models
1.3.3. Gaussian Linear State-Space Models
1.3.4. Conditionally Gaussian Linear State-Space Models
1.3.5. General (Continuous) State-Space HMMs
1.3.6. Switching Processes with Markov Regime
1.4. Left-to-Right and Ergodic Hidden Markov Models
2. Main Definitions and Notations
2.1. Markov Chains
2.1.1. Transition Kernels
2.1.2. Homogeneous Markov Chains
2.1.3. Non-homogeneous Markov Chains
2.2. Hidden Markov Models
2.2.1. Definitions and Notations
2.2.2. Conditional Independence in Hidden Markov Models
2.2.3. Hierarchical Hidden Markov Models
Part I. State Inference
3. Filtering and Smoothing Recursions
3.1. Basic Notations and Definitions
3.1.1. Likelihood
3.1.2. Smoothing
3.1.3. The Forward-Backward Decomposition
3.1.4. Implicit Conditioning (Please Read This Section!)
3.2. Forward-Backward
3.2.1. The Forward-Backward Recursions
3.2.2. Filtering and Normalized Recursion
3.3. Markovian Decompositions
3.3.1. Forward Decomposition
3.3.2. Backward Decomposition
3.4. Complements
4. Advanced Topics in Smoothing
4.1. Recursive Computation of Smoothed Functionals
4.1.1. Fixed Point Smoothing
4.1.2. Recursive Smoothers for General Functionals
4.1.3. Comparison with Forward-Backward Smoothing
4.2. Filtering and Smoothing in More General Models
4.2.1. Smoothing in Markov-switching Models
4.2.2. Smoothing in Partially Observed Markov Chains
4.2.3. Marginal Smoothing in Hierarchical HMMs
4.3. Forgetting of the Initial Condition
4.3.1. Total Variation
4.3.2. Lipshitz Contraction for Transition Kernels
4.3.3. The Doeblin Condition and Uniform Ergodicity
4.3.4. Forgetting Properties
4.3.5. Uniform Forgetting Under Strong Mixing Conditions
4.3.6. Forgetting Under Alternative Conditions
5. Applications of Smoothing
5.1. Models with Finite State Space
5.1.1. Smoothing
5.1.2. Maximum a Posteriori Sequence Estimation
5.2. Gaussian Linear State-Space Models
5.2.1. Filtering and Backward Markovian Smoothing
5.2.2. Linear Prediction Interpretation
5.2.3. The Prediction and Filtering Recursions Revisited
5.2.4. Disturbance Smoothing
5.2.5. The Backward Recursion and the Two-Filter Formula
5.2.6. Application to Marginal Filtering and Smoothing in CGLSSMs
6. Monte Carlo Methods
6.1. Basic Monte Carlo Methods
6.1.1. Monte Carlo Integration
6.1.2. Monte Carlo Simulation for HMM State Inference
6.2. A Markov Chain Monte Carlo Primer
6.2.1. The Accept-Reject Algorithm
6.2.2. Markov Chain Monte Carlo
6.2.3. Metropolis-Hastings
6.2.4. Hybrid Algorithms
6.2.5. Gibbs Sampling
6.2.6. Stopping an MCMC Algorithm
6.3. Applications to Hidden Markov Models
6.3.1. Generic Sampling Strategies
6.3.2. Gibbs Sampling in CGLSSMs
7. Sequential Monte Carlo Methods
7.1. Importance Sampling and Resampling
7.1.1. Importance Sampling
7.1.2. Sampling Importance Resampling
7.2. Sequential Importance Sampling
7.2.1. Sequential Implementation for HMMs
7.2.2. Choice of the Instrumental Kernel
7.3. Sequential Improtance Sampling with Resampling
7.3.1. Weight Degeneracy
7.3.2. Resampling
7.4. Complements
7.4.1. Implementation of Multinomial Resampling
7.4.2. Alternatives to Multinomial Resampling
8. Advanced Topics in Sequential Monte Carlo
8.1. Alternatives to SISR
8.1.1. I.I.D. Sampling
8.1.2. Two-Stage Sampling
8.1.3. Interpretation with Auxiliary Variables
8.1.4. Auxiliary Accept-Reject Sampling
8.1.5. Markov Chain Monte Carlo Auxiliary Sampling
8.2. Sequential Monte Carlo in Hierarchical HMMs
8.2.1. Sequential Importance Sampling and Global Sampling
8.2.2. Optimal Sampling
8.2.3. Application to CGLSSMs
8.3. Particle Approximation of Smoothing Functionals
9. Analysis of Sequential Monte Carlo Methods
9.1. Importance Sampling
9.1.1. Unnormalized Importance Sampling
9.1.2. Deviation Inequalities
9.1.3. Self-normalized Importance Sampling Estimator
9.2. Sampling Importance Resampling
9.2.1. The Algorithm
9.2.2. Definitions and Notations
9.2.3. Weighting and Resampling
9.2.4. Application to the Single-Stage SIR Algorithm
9.3. Single-Step Analysis of SMC Methods
9.3.1. Mutation Step
9.3.2. Description of Algorithms
9.3.3. Analysis of the Mutation/Selection Algorithm
9.3.4. Analysis of the Selection/Mutation Algorithm
9.4. Sequential Monte Carlo Methods
9.4.1. SISR
9.4.2. I.I.D. Sampling
9.5. Complements
9.5.1. Weak Limits Theorems for Triangular Array
9.5.2. Bibliographic Notes
Part II. Parameter Inference
10. Maximum Likelihood Inference, Part I: Optimization Through Exact Smoothing
10.1. Likelihood Optimization in Incomplete Data Models
10.1.1. Problem Statement and Notations
10.1.2. The Expectation-Maximization Algorithm
10.1.3. Gradient-based Methods
10.1.4. Pros and Cons of Gradient-based Methods
10.2. Application to HMMs
10.2.1. Hidden Markov Models as Missing Data Models
10.2.2. EM in HMMs
10.2.3. Computing Derivatives
10.2.4. Connection with the Sensitivity Equation Approach
10.3. The Example of Normal Hidden Markov Models
10.3.1. EM Parameter Update Formulas
10.3.2. Estimation of the Initial Distribution
10.3.3. Recursive Implementation of E-Step
10.3.4. Computation of the Score and Observed Information
10.4. The Example of Gaussian Linear State-Space Models
10.4.1. The Intermediate Quantity of EM
10.4.2. Recursive Implementation
10.5. Complements
10.5.1. Global Convergence of the EM Algorithm
10.5.2. Rate of Convergence of EM
10.5.3. Generalized EM Algorithms
10.5.4. Bibliographic Notes
11. Maximum Likelihood Inference, Part II: Monte Carlo Optimization
11.1. Methods and Algorithms
11.1.1. Monte Carlo EM
11.1.2. Simulation Schedules
11.1.3. Gradient-based Algorithms
11.1.4. Interlude: Stochastic Approximation and the Robbins-Monro Approach
11.1.5. Stochastic Gradient Algorithms
11.1.6. Stochastic Approximation EM
11.1.7. Stochastic EM
11.2. Analysis of the MCEM Algorithm
11.2.1. Convergence of Perturbed Dynamical Systems
11.2.2. Convergence of the MCEM Algorithm
11.2.3. Rate of Convergence of MCEM
11.3. Analysis of Stochastic Approximation Algorithms
11.3.1. Basic Results for Stochastic Approximation Algorithms
11.3.2. Convergence of the Stochastic Gradient Algorithm
11.3.3. Rate of Convergence of the Stochastic Gradient Algorithm
11.3.4. Convergence of the SAEM Algorithm
11.4. Complements
12. Statistical Properties of the Maximum Likelihood Estimator
12.1. A Primer on MLE Asymptotics
12.2. Stationary Approximations
12.3. Consistency
12.3.1. Construction of the Stationary Conditional Log-likelihood
12.3.2. The Contrast Function and Its Properties
12.4. Identifiability
12.4.1. Equivalence of Parameters
12.4.2. Identifiability of Mixture Densities
12.4.3. Application of Mixture Identifiability to Hidden Markov Models
12.5. Asymptotic Normality of the Score and Convergence of the Observed Information
12.5.1. The Score Function and Invoking the Fisher Identity
12.5.2. Construction of the Stationary Conditional Score
12.5.3. Weak Convergence of the Normalized Score
12.5.4. Convergence of the Normalized Observed Information
12.5.5. Asymptotics of the Maximum Likelihood Estimator
12.6. Applications to Likelihood-based Tests
12.7. Complements
13. Fully Bayesian Approaches
13.1. Parameter Estimation
13.1.1. Bayesian Inference
13.1.2. Prior Distributions for HMMs
13.1.3. Non-identifiability and Label Switching
13.1.4. MCMC Methods for Bayesian Inference
13.2. Reversible Jump Methods
13.2.1. Variable Dimension Models
13.2.2. Green's Reversible Jump Algorithm
13.2.3. Alternative Sampler Designs
13.2.4. Alternatives to Reversible Jump MCMC
13.3. Multiple Imputations Methods and Maximum a Posteriori
13.3.1. Simulated Annealing
13.3.2. The SAME Algorithm
Part III. Background and Complements
14. Elements of Markov Chain Theory
14.1. Chains on Countable State Spaces
14.1.1. Irreducibility
14.1.2. Recurrence and Transience
14.1.3. Invariant Measures and Stationarity
14.1.4. Ergodicity
14.2. Chains on General State Spaces
14.2.1. Irreducibility
14.2.2. Recurrence and Transience
14.2.3. Invariant Measures and Stationarity
14.2.4. Ergodicity
14.2.5. Geometric Ergodicity and Foster-Lyapunov Conditions
14.2.6. Limit Theorems
14.3. Applications to Hidden Markov Models
14.3.1. Phi-irreducibility
14.3.2. Atoms and Small Sets
14.3.3. Recurrence and Positive Recurrence
15. An Information-Theoretic Perspective on Order Estimation
15.1. Model Order Identification: What Is It About?
15.2. Order Estimation in Perspective
15.3. Order Estimation and Composite Hypothesis Testing
15.4. Code-based Identification
15.4.1. Definitions
15.4.2. Information Divergence Rates
15.5. MDL Order Estimators in Bayesian Settings
15.6. Strongly Consistent Penalized Maximum Likelihood Estimators for HMM Order Estimation
15.7. Efficiency Issues
15.7.1. Variations on Stein's Lemma
15.7.2. Achieving Optimal Error Exponents
15.8. Consistency of the BIC Estimator in the Markov Order Estimation Problem
15.8.1. Some Martingale Tools
15.8.2. The Martingale Approach
15.8.3. The Union Bound Meets Martingale Inequalities
15.9. Complements
Part IV. Appendices
A. Conditioning
A.1. Probability and Topology Terminology and Notation
A.2. Conditional Expectation
A.3. Conditional Distribution
A.4. Conditional Independence
B. Linear Prediction
B.1. Hilbert Spaces
B.2. The Projection Theorem
C. Notations
C.1. Mathematical
C.2. Probability
C.3. Hidden Markov Models
C.4. Sequential Monte Carlo
References
Index

Inference in hidden Markov models /

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