# Factorial hidden Markov models

August 29, 2022 — August 29, 2022

Bayes

dynamical systems

linear algebra

probability

signal processing

state space models

statistics

time series

Placeholder. Factorial HMMs factorize hidden state into independent (?) state variables. This obviously-nuts assumption makes them tractable but still expressive, which is what works for neural nets, so I guess we are cool.

## 1 References

Aasnaes, and Kailath. 1973. “An Innovations Approach to Least-Squares Estimation–Part VII: Some Applications of Vector Autoregressive-Moving Average Models.”

*IEEE Transactions on Automatic Control*.
Alliney. 1992. “Digital Filters as Absolute Norm Regularizers.”

*IEEE Transactions on Signal Processing*.
Alzraiee, White, Knowling, et al. 2022. “A Scalable Model-Independent Iterative Data Assimilation Tool for Sequential and Batch Estimation of High Dimensional Model Parameters and States.”

*Environmental Modelling & Software*.
Ansley, and Kohn. 1985. “Estimation, Filtering, and Smoothing in State Space Models with Incompletely Specified Initial Conditions.”

*The Annals of Statistics*.
Arulampalam, Maskell, Gordon, et al. 2002. “A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking.”

*IEEE Transactions on Signal Processing*.
Bagge Carlson. 2018. “Machine Learning and System Identification for Estimation in Physical Systems.”

Battey, and Sancetta. 2013. “Conditional Estimation for Dependent Functional Data.”

*Journal of Multivariate Analysis*.
Batz, Ruttor, and Opper. 2017. “Approximate Bayes Learning of Stochastic Differential Equations.”

*arXiv:1702.05390 [Physics, Stat]*.
Becker, Pandya, Gebhardt, et al. 2019. “Recurrent Kalman Networks: Factorized Inference in High-Dimensional Deep Feature Spaces.” In

*International Conference on Machine Learning*.
Berkhout, and Zaanen. 1976. “A Comparison Between Wiener Filtering, Kalman Filtering, and Deterministic Least Squares Estimation*.”

*Geophysical Prospecting*.
Bilmes. 1998. “A Gentle Tutorial of the EM Algorithm and Its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models.”

*International Computer Science Institute*.
Bishop, and Del Moral. 2016. “On the Stability of Kalman-Bucy Diffusion Processes.”

*SIAM Journal on Control and Optimization*.
———. 2023. “On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering.”

*Mathematics of Control, Signals, and Systems*.
Bishop, Del Moral, and Pathiraja. 2017. “Perturbations and Projections of Kalman-Bucy Semigroups Motivated by Methods in Data Assimilation.”

*arXiv:1701.05978 [Math]*.
Bretó, He, Ionides, et al. 2009. “Time Series Analysis via Mechanistic Models.”

*The Annals of Applied Statistics*.
Brunton, Proctor, and Kutz. 2016. “Discovering Governing Equations from Data by Sparse Identification of Nonlinear Dynamical Systems.”

*Proceedings of the National Academy of Sciences*.
Campbell, Shi, Rainforth, et al. 2021. “Online Variational Filtering and Parameter Learning.” In.

Carmi. 2013. “Compressive System Identification: Sequential Methods and Entropy Bounds.”

*Digital Signal Processing*.
———. 2014. “Compressive System Identification.” In

*Compressed Sensing & Sparse Filtering*. Signals and Communication Technology.
Cassidy, Rae, and Solo. 2015. “Brain Activity: Connectivity, Sparsity, and Mutual Information.”

*IEEE Transactions on Medical Imaging*.
Cauchemez, and Ferguson. 2008. “Likelihood-Based Estimation of Continuous-Time Epidemic Models from Time-Series Data: Application to Measles Transmission in London.”

*Journal of The Royal Society Interface*.
Charles, Balavoine, and Rozell. 2016. “Dynamic Filtering of Time-Varying Sparse Signals via L1 Minimization.”

*IEEE Transactions on Signal Processing*.
Chen, Y., and Hero. 2012. “Recursive ℓ1,∞ Group Lasso.”

*IEEE Transactions on Signal Processing*.
Chen, Bin, and Hong. 2012. “Testing for the Markov Property in Time Series.”

*Econometric Theory*.
Chung, Kastner, Dinh, et al. 2015. “A Recurrent Latent Variable Model for Sequential Data.” In

*Advances in Neural Information Processing Systems 28*.
Clark, and Bjørnstad. 2004. “Population Time Series: Process Variability, Observation Errors, Missing Values, Lags, and Hidden States.”

*Ecology*.
Commandeur, and Koopman. 2007.

*An Introduction to State Space Time Series Analysis*.
Cox, van de Laar, and de Vries. 2019. “A Factor Graph Approach to Automated Design of Bayesian Signal Processing Algorithms.”

*International Journal of Approximate Reasoning*.
Cressie, and Huang. 1999. “Classes of Nonseparable, Spatio-Temporal Stationary Covariance Functions.”

*Journal of the American Statistical Association*.
Cressie, Shi, and Kang. 2010. “Fixed Rank Filtering for Spatio-Temporal Data.”

*Journal of Computational and Graphical Statistics*.
Cressie, and Wikle. 2011.

*Statistics for Spatio-Temporal Data*. Wiley Series in Probability and Statistics 2.0.
Freitas, João FG de, Doucet, Niranjan, et al. 1998. “Global Optimisation of Neural Network Models via Sequential Sampling.” In

*Proceedings of the 11th International Conference on Neural Information Processing Systems*. NIPS’98.
Freitas, J. F. G. de, Niranjan, Gee, et al. 1998. “Sequential Monte Carlo Methods for Optimisation of Neural Network Models.”

*Cambridge University Engineering Department, Cambridge, England, Technical Report TR-328*.
Deisenroth, and Mohamed. 2012. “Expectation Propagation in Gaussian Process Dynamical Systems.” In

*Proceedings of the 25th International Conference on Neural Information Processing Systems - Volume 2*. NIPS’12.
Del Moral, Kurtzmann, and Tugaut. 2017. “On the Stability and the Uniform Propagation of Chaos of a Class of Extended Ensemble Kalman-Bucy Filters.”

*SIAM Journal on Control and Optimization*.
Doucet, Jacob, and Rubenthaler. 2013. “Derivative-Free Estimation of the Score Vector and Observed Information Matrix with Application to State-Space Models.”

*arXiv:1304.5768 [Stat]*.
Durbin, and Koopman. 1997. “Monte Carlo Maximum Likelihood Estimation for Non-Gaussian State Space Models.”

*Biometrika*.
———. 2012.

*Time Series Analysis by State Space Methods*. Oxford Statistical Science Series 38.
Duttweiler, and Kailath. 1973a. “RKHS Approach to Detection and Estimation Problems–IV: Non-Gaussian Detection.”

*IEEE Transactions on Information Theory*.
———. 1973b. “RKHS Approach to Detection and Estimation Problems–V: Parameter Estimation.”

*IEEE Transactions on Information Theory*.
Easley, and Berry. 2020. “A Higher Order Unscented Transform.”

*arXiv:2006.13429 [Cs, Math]*.
Eddy. 1996. “Hidden Markov Models.”

*Current Opinion in Structural Biology*.
Eden, Frank, Barbieri, et al. 2004. “Dynamic Analysis of Neural Encoding by Point Process Adaptive Filtering.”

*Neural Computation*.
Edwards, and Ankinakatte. 2015. “Context-Specific Graphical Models for Discrete Longitudinal Data.”

*Statistical Modelling*.
Eleftheriadis, Nicholson, Deisenroth, et al. 2017. “Identification of Gaussian Process State Space Models.” In

*Advances in Neural Information Processing Systems 30*.
Fearnhead, and Künsch. 2018. “Particle Filters and Data Assimilation.”

*Annual Review of Statistics and Its Application*.
Finke, and Singh. 2016. “Approximate Smoothing and Parameter Estimation in High-Dimensional State-Space Models.”

*arXiv:1606.08650 [Stat]*.
Föll, Haasdonk, Hanselmann, et al. 2017. “Deep Recurrent Gaussian Process with Variational Sparse Spectrum Approximation.”

*arXiv:1711.00799 [Stat]*.
Fraccaro, Sø nderby, Paquet, et al. 2016. “Sequential Neural Models with Stochastic Layers.” In

*Advances in Neural Information Processing Systems 29*.
Fraser. 2008.

*Hidden Markov Models and Dynamical Systems*.
Friedlander, Kailath, and Ljung. 1975. “Scattering Theory and Linear Least Squares Estimation: Part II: Discrete-Time Problems.” In

*1975 IEEE Conference on Decision and Control Including the 14th Symposium on Adaptive Processes*.
Frigola, Chen, and Rasmussen. 2014. “Variational Gaussian Process State-Space Models.” In

*Advances in Neural Information Processing Systems 27*.
Frigola, Lindsten, Schön, et al. 2013. “Bayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC.” In

*Advances in Neural Information Processing Systems 26*.
Friston. 2008. “Variational Filtering.”

*NeuroImage*.
Gevers, and Kailath. 1973. “An Innovations Approach to Least-Squares Estimation–Part VI: Discrete-Time Innovations Representations and Recursive Estimation.”

*IEEE Transactions on Automatic Control*.
Gorad, Zhao, and Särkkä. 2020. “Parameter Estimation in Non-Linear State-Space Models by Automatic Differentiation of Non-Linear Kalman Filters.” In.

Gottwald, and Reich. 2020. “Supervised Learning from Noisy Observations: Combining Machine-Learning Techniques with Data Assimilation.”

*arXiv:2007.07383 [Physics, Stat]*.
Gourieroux, and Jasiak. 2015. “Filtering, Prediction and Simulation Methods for Noncausal Processes.”

*Journal of Time Series Analysis*.
Gu, Johnson, Goel, et al. 2021. “Combining Recurrent, Convolutional, and Continuous-Time Models with Linear State Space Layers.” In

*Advances in Neural Information Processing Systems*.
Haber, Lucka, and Ruthotto. 2018. “Never Look Back - A Modified EnKF Method and Its Application to the Training of Neural Networks Without Back Propagation.”

*arXiv:1805.08034 [Cs, Math]*.
Hamilton, Berry, and Sauer. 2016. “Kalman-Takens Filtering in the Presence of Dynamical Noise.”

*arXiv:1611.05414 [Physics, Stat]*.
Hartikainen, and Särkkä. 2010. “Kalman Filtering and Smoothing Solutions to Temporal Gaussian Process Regression Models.” In

*2010 IEEE International Workshop on Machine Learning for Signal Processing*.
Harvey, A., and Koopman. 2005. “Structural Time Series Models.” In

*Encyclopedia of Biostatistics*.
Harvey, Andrew, and Luati. 2014. “Filtering With Heavy Tails.”

*Journal of the American Statistical Association*.
Hefny, Downey, and Gordon. 2015. “A New View of Predictive State Methods for Dynamical System Learning.”

*arXiv:1505.05310 [Cs, Stat]*.
He, Ionides, and King. 2010. “Plug-and-Play Inference for Disease Dynamics: Measles in Large and Small Populations as a Case Study.”

*Journal of The Royal Society Interface*.
Hong, Mitchell, Chen, et al. 2008. “Model Selection Approaches for Non-Linear System Identification: A Review.”

*International Journal of Systems Science*.
Hou, Lawrence, and Hero. 2016. “Penalized Ensemble Kalman Filters for High Dimensional Non-Linear Systems.”

*arXiv:1610.00195 [Physics, Stat]*.
Hsiao, and Schultz. 2011. “Generalized Baum-Welch Algorithm and Its Implication to a New Extended Baum-Welch Algorithm.” In

*In Proceedings of INTERSPEECH*.
Hsu, Kakade, and Zhang. 2012. “A Spectral Algorithm for Learning Hidden Markov Models.”

*Journal of Computer and System Sciences*, JCSS Special Issue: Cloud Computing 2011,.
Huber. 2014. “Recursive Gaussian Process: On-Line Regression and Learning.”

*Pattern Recognition Letters*.
Ionides, Edward L., Bhadra, Atchadé, et al. 2011. “Iterated Filtering.”

*The Annals of Statistics*.
Ionides, E. L., Bretó, and King. 2006. “Inference for Nonlinear Dynamical Systems.”

*Proceedings of the National Academy of Sciences*.
Johansen, Doucet, and Davy. 2006. “Sequential Monte Carlo for Marginal Optimisation Problems.”

*Scis & Isis*.
Johnson. 2012. “A Simple Explanation of A Spectral Algorithm for Learning Hidden Markov Models.”

*arXiv:1204.2477 [Cs, Stat]*.
Julier, Uhlmann, and Durrant-Whyte. 1995. “A New Approach for Filtering Nonlinear Systems.” In

*American Control Conference, Proceedings of the 1995*.
Kailath. 1971. “RKHS Approach to Detection and Estimation Problems–I: Deterministic Signals in Gaussian Noise.”

*IEEE Transactions on Information Theory*.
———. 1974. “A View of Three Decades of Linear Filtering Theory.”

*IEEE Transactions on Information Theory*.
Kailath, and Duttweiler. 1972. “An RKHS Approach to Detection and Estimation Problems– III: Generalized Innovations Representations and a Likelihood-Ratio Formula.”

*IEEE Transactions on Information Theory*.
Kailath, and Geesey. 1971. “An Innovations Approach to Least Squares Estimation–Part IV: Recursive Estimation Given Lumped Covariance Functions.”

*IEEE Transactions on Automatic Control*.
———. 1973. “An Innovations Approach to Least-Squares Estimation–Part V: Innovations Representations and Recursive Estimation in Colored Noise.”

*IEEE Transactions on Automatic Control*.
Kailath, and Weinert. 1975. “An RKHS Approach to Detection and Estimation Problems–II: Gaussian Signal Detection.”

*IEEE Transactions on Information Theory*.
Kalman, R. 1959. “On the General Theory of Control Systems.”

*IRE Transactions on Automatic Control*.
Kalman, R. E. 1960. “A New Approach to Linear Filtering and Prediction Problems.”

*Journal of Basic Engineering*.
Kalouptsidis, Mileounis, Babadi, et al. 2011. “Adaptive Algorithms for Sparse System Identification.”

*Signal Processing*.
Karvonen, and Särkkä. 2016. “Approximate State-Space Gaussian Processes via Spectral Transformation.” In

*2016 IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP)*.
Kelly, Law, and Stuart. 2014. “Well-Posedness and Accuracy of the Ensemble Kalman Filter in Discrete and Continuous Time.”

*Nonlinearity*.
Kirch, Edwards, Meier, et al. 2019. “Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis.”

*Bayesian Analysis*.
Kitagawa. 1987. “Non-Gaussian State—Space Modeling of Nonstationary Time Series.”

*Journal of the American Statistical Association*.
———. 1996. “Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models.”

*Journal of Computational and Graphical Statistics*.
Kitagawa, and Gersch. 1996.

*Smoothness Priors Analysis of Time Series*. Lecture notes in statistics 116.
Kobayashi, Mark, and Turin. 2011.

*Probability, Random Processes, and Statistical Analysis: Applications to Communications, Signal Processing, Queueing Theory and Mathematical Finance*.
Koopman, and Durbin. 2000. “Fast Filtering and Smoothing for Multivariate State Space Models.”

*Journal of Time Series Analysis*.
Krishnan, Shalit, and Sontag. 2017. “Structured Inference Networks for Nonlinear State Space Models.” In

*Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence*.
Kulhavý. 1990. “Recursive Nonlinear Estimation: A Geometric Approach.”

*Automatica*.
———. 1996.

*Recursive Nonlinear Estimation*. Lecture Notes in Control and Information Sciences.
Kutschireiter, Surace, Sprekeler, et al. 2015. “Approximate Nonlinear Filtering with a Recurrent Neural Network.”

*BMC Neuroscience*.
Lázaro-Gredilla, Quiñonero-Candela, Rasmussen, et al. 2010. “Sparse Spectrum Gaussian Process Regression.”

*Journal of Machine Learning Research*.
Le Gland, Monbet, and Tran. 2009. “Large Sample Asymptotics for the Ensemble Kalman Filter.” Report.

Lei, Bickel, and Snyder. 2009. “Comparison of Ensemble Kalman Filters Under Non-Gaussianity.”

*Monthly Weather Review*.
Levin. 2017. “The Inner Structure of Time-Dependent Signals.”

*arXiv:1703.08596 [Cs, Math, Stat]*.
Lindgren, Rue, and Lindström. 2011. “An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach.”

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.
Ljung, and Kailath. 1976. “Backwards Markovian Models for Second-Order Stochastic Processes (Corresp.).”

*IEEE Transactions on Information Theory*.
Ljung, Kailath, and Friedlander. 1975. “Scattering Theory and Linear Least Squares Estimation: Part I: Continuous-Time Problems.” In

*1975 IEEE Conference on Decision and Control Including the 14th Symposium on Adaptive Processes*.
Loeliger, Dauwels, Hu, et al. 2007. “The Factor Graph Approach to Model-Based Signal Processing.”

*Proceedings of the IEEE*.
Manton, Krishnamurthy, and Poor. 1998. “James-Stein State Filtering Algorithms.”

*IEEE Transactions on Signal Processing*.
Mattos, Dai, Damianou, et al. 2016. “Recurrent Gaussian Processes.” In

*Proceedings of ICLR*.
Mattos, Dai, Damianou, et al. 2017. “Deep Recurrent Gaussian Processes for Outlier-Robust System Identification.”

*Journal of Process Control*, DYCOPS-CAB 2016,.
Meyer, Edwards, Maturana-Russel, et al. 2020. “Computational Techniques for Parameter Estimation of Gravitational Wave Signals.”

*WIREs Computational Statistics*.
Micchelli, and Olsen. 2000. “Penalized Maximum-Likelihood Estimation, the Baum–Welch Algorithm, Diagonal Balancing of Symmetric Matrices and Applications to Training Acoustic Data.”

*Journal of Computational and Applied Mathematics*.
Miller, Glennie, and Seaton. 2020. “Understanding the Stochastic Partial Differential Equation Approach to Smoothing.”

*Journal of Agricultural, Biological and Environmental Statistics*.
Nickisch, Solin, and Grigorevskiy. 2018. “State Space Gaussian Processes with Non-Gaussian Likelihood.” In

*International Conference on Machine Learning*.
Olfati-Saber. 2005. “Distributed Kalman Filter with Embedded Consensus Filters.” In

*44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC ’05*.
Ollivier. 2017. “Online Natural Gradient as a Kalman Filter.”

*arXiv:1703.00209 [Math, Stat]*.
Papadopoulos, Pachet, Roy, et al. 2015. “Exact Sampling for Regular and Markov Constraints with Belief Propagation.” In

*Principles and Practice of Constraint Programming*. Lecture Notes in Computer Science.
Perry. 2010. “Andrew Viterbi’s Fabulous Formula [Medal of Honor].”

*IEEE Spectrum*.
Picci. 1991. “Stochastic Realization Theory.” In

*Mathematical System Theory: The Influence of R. E. Kalman*.
Psiaki. 2013. “The Blind Tricyclist Problem and a Comparative Study of Nonlinear Filters: A Challenging Benchmark for Evaluating Nonlinear Estimation Methods.”

*IEEE Control Systems*.
Pugachev, V.S. 1982. “Conditionally Optimal Estimation in Stochastic Differential Systems.”

*Automatica*.
Pugachev, V. S., and Sinit︠s︡yn. 2001.

*Stochastic systems: theory and applications*.
Quiñonero-Candela, and Rasmussen. 2005. “A Unifying View of Sparse Approximate Gaussian Process Regression.”

*Journal of Machine Learning Research*.
Rabiner, L.R. 1989. “A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition.”

*Proceedings of the IEEE*.
Rabiner, L., and Juang. 1986. “An Introduction to Hidden Markov Models.”

*IEEE ASSP Magazine*.
Raol, and Sinha. 1987. “On Pugachev’s Filtering Theory for Stochastic Nonlinear Systems.” In

*Stochastic Control*. IFAC Symposia Series.
Reece, and Roberts. 2010. “An Introduction to Gaussian Processes for the Kalman Filter Expert.” In

*2010 13th International Conference on Information Fusion*.
Reller. 2013. “State-Space Methods in Statistical Signal Processing: New Ideas and Applications.” Application/pdf.

Revach, Shlezinger, van Sloun, et al. 2021. “Kalmannet: Data-Driven Kalman Filtering.” In

*ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*.
Robertson, Andrew N. 2011. “A Bayesian Approach to Drum Tracking.” In.

Robertson, Andrew, and Plumbley. 2007. “B-Keeper: A Beat-Tracker for Live Performance.” In

*Proceedings of the 7th International Conference on New Interfaces for Musical Expression*. NIME ’07.
Robertson, Andrew, Stark, and Davies. 2013. “Percussive Beat Tracking Using Real-Time Median Filtering.” In

*Proceedings of European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases*.
Robertson, Andrew, Stark, and Plumbley. 2011. “Real-Time Visual Beat Tracking Using a Comb Filter Matrix.” In

*Proceedings of the International Computer Music Conference 2011*.
Rodriguez, and Ruiz. 2009. “Bootstrap Prediction Intervals in State–Space Models.”

*Journal of Time Series Analysis*.
Roth, Hendeby, Fritsche, et al. 2017. “The Ensemble Kalman Filter: A Signal Processing Perspective.”

*EURASIP Journal on Advances in Signal Processing*.
Rozet, and Louppe. 2023. “Score-Based Data Assimilation.”

Rudenko. 2013. “Optimal Structure of Continuous Nonlinear Reduced-Order Pugachev Filter.”

*Journal of Computer and Systems Sciences International*.
Särkkä, Simo. 2007. “On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems.”

*IEEE Transactions on Automatic Control*.
———. 2013.

*Bayesian Filtering and Smoothing*. Institute of Mathematical Statistics Textbooks 3.
Särkkä, Simo, and Hartikainen. 2012. “Infinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression.” In

*Artificial Intelligence and Statistics*.
Särkkä, S., and Hartikainen. 2013. “Non-Linear Noise Adaptive Kalman Filtering via Variational Bayes.” In

*2013 IEEE International Workshop on Machine Learning for Signal Processing (MLSP)*.
Särkkä, Simo, and Nummenmaa. 2009. “Recursive Noise Adaptive Kalman Filtering by Variational Bayesian Approximations.”

*IEEE Transactions on Automatic Control*.
Särkkä, Simo, Solin, and Hartikainen. 2013. “Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering.”

*IEEE Signal Processing Magazine*.
Schein, Wallach, and Zhou. 2016. “Poisson-Gamma Dynamical Systems.” In

*Advances In Neural Information Processing Systems*.
Schmidt, Krämer, and Hennig. 2021. “A Probabilistic State Space Model for Joint Inference from Differential Equations and Data.”

*arXiv:2103.10153 [Cs, Stat]*.
Segall, Davis, and Kailath. 1975. “Nonlinear Filtering with Counting Observations.”

*IEEE Transactions on Information Theory*.
Šindelář, Vajda, and Kárnỳ. 2008. “Stochastic Control Optimal in the Kullback Sense.”

*Kybernetika*.
Sorenson. 1970. “Least-Squares Estimation: From Gauss to Kalman.”

*IEEE Spectrum*.
Städler, and Mukherjee. 2013. “Penalized Estimation in High-Dimensional Hidden Markov Models with State-Specific Graphical Models.”

*The Annals of Applied Statistics*.
Surace, and Pfister. 2016. “Online Maximum Likelihood Estimation of the Parameters of Partially Observed Diffusion Processes.” In.

Tavakoli, and Panaretos. 2016. “Detecting and Localizing Differences in Functional Time Series Dynamics: A Case Study in Molecular Biophysics.”

*Journal of the American Statistical Association*.
Thrun, and Langford. 1998. “Monte Carlo Hidden Markov Models.”

Thrun, Langford, and Fox. 1999. “Monte Carlo Hidden Markov Models: Learning Non-Parametric Models of Partially Observable Stochastic Processes.” In

*Proceedings of the International Conference on Machine Learning*.
Turner, Deisenroth, and Rasmussen. 2010. “State-Space Inference and Learning with Gaussian Processes.” In

*Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics*.
Wikle, and Berliner. 2007. “A Bayesian Tutorial for Data Assimilation.”

*Physica D: Nonlinear Phenomena*, Data Assimilation,.
Wikle, Berliner, and Cressie. 1998. “Hierarchical Bayesian Space-Time Models.”

*Environmental and Ecological Statistics*.
Zhao, and Cui. 2023. “Tensor-Based Methods for Sequential State and Parameter Estimation in State Space Models.”

Zoeter. 2007. “Bayesian Generalized Linear Models in a Terabyte World.” In

*2007 5th International Symposium on Image and Signal Processing and Analysis*.