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