The chapter provides a comprehensive review of linear Gaussian models, including factor analysis, principal component analysis (PCA), mixtures of Gaussian clusters, vector quantization, Kalman filter models, and hidden Markov models (HMMs). These models are unified under a single generative framework, where the state of the process is represented by a vector \( \mathbf{x} \) that evolves according to a first-order Markov dynamics, and the output is generated from the current state through a linear observation process. Both state evolution and observation processes are corrupted by additive Gaussian noise. The authors introduce a new model called Sensible Principal Component Analysis (SPCA) and a concept of spatially adaptive observation noise. They also review global and local mixtures of the basic models and provide pseudocode for inference and learning. Key points include: - Factor analysis and mixtures of Gaussians can be implemented using autoencoder neural networks with different nonlinearities. - Independent component analysis (ICA) is a nonlinear version of factor analysis. - SPCA is a static data model based on PCA but with a probabilistic interpretation. - The EM algorithm is used for learning in these models, maximizing the likelihood of the observed data given the hidden states. The chapter also discusses the relationship between continuous and discrete state models, and how these models can be extended to handle time-series data and discrete states.The chapter provides a comprehensive review of linear Gaussian models, including factor analysis, principal component analysis (PCA), mixtures of Gaussian clusters, vector quantization, Kalman filter models, and hidden Markov models (HMMs). These models are unified under a single generative framework, where the state of the process is represented by a vector \( \mathbf{x} \) that evolves according to a first-order Markov dynamics, and the output is generated from the current state through a linear observation process. Both state evolution and observation processes are corrupted by additive Gaussian noise. The authors introduce a new model called Sensible Principal Component Analysis (SPCA) and a concept of spatially adaptive observation noise. They also review global and local mixtures of the basic models and provide pseudocode for inference and learning. Key points include: - Factor analysis and mixtures of Gaussians can be implemented using autoencoder neural networks with different nonlinearities. - Independent component analysis (ICA) is a nonlinear version of factor analysis. - SPCA is a static data model based on PCA but with a probabilistic interpretation. - The EM algorithm is used for learning in these models, maximizing the likelihood of the observed data given the hidden states. The chapter also discusses the relationship between continuous and discrete state models, and how these models can be extended to handle time-series data and discrete states.