This paper introduces a probabilistic approach to principal component analysis (PCA), demonstrating how PCA can be derived from a Gaussian latent variable model closely related to factor analysis. The authors show that the principal axes of data can be determined through maximum likelihood estimation of parameters in this model, leading to an EM algorithm for estimating the principal subspace. The probabilistic formulation of PCA offers advantages such as the ability to extend conventional PCA, handle missing data, and integrate with Bayesian methods. It also allows for the use of probabilistic PCA as a constrained Gaussian density model, enabling efficient parameter estimation and applications in classification and novelty detection. The paper discusses the properties of the likelihood function, the EM algorithm for PCA, and the relationship between PCA and factor analysis. It also provides examples of practical applications, including data visualization with missing values, mixture models for complex data structures, and controlling model complexity through the choice of latent space dimension. The paper concludes that probabilistic PCA provides a flexible and powerful framework for dimensionality reduction and data modeling.This paper introduces a probabilistic approach to principal component analysis (PCA), demonstrating how PCA can be derived from a Gaussian latent variable model closely related to factor analysis. The authors show that the principal axes of data can be determined through maximum likelihood estimation of parameters in this model, leading to an EM algorithm for estimating the principal subspace. The probabilistic formulation of PCA offers advantages such as the ability to extend conventional PCA, handle missing data, and integrate with Bayesian methods. It also allows for the use of probabilistic PCA as a constrained Gaussian density model, enabling efficient parameter estimation and applications in classification and novelty detection. The paper discusses the properties of the likelihood function, the EM algorithm for PCA, and the relationship between PCA and factor analysis. It also provides examples of practical applications, including data visualization with missing values, mixture models for complex data structures, and controlling model complexity through the choice of latent space dimension. The paper concludes that probabilistic PCA provides a flexible and powerful framework for dimensionality reduction and data modeling.