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This is a tutorial and survey paper on factor analysis, probabilisticPrincipal Component Analysis (PCA), variational inference, and VariationalAutoencoder (VAE). These methods, which are tightly related, are dimensionalityreduction and generative models. They assume that every data point is generatedfrom or caused by a low-dimensional latent factor. By learning the parametersof distribution of latent space, the corresponding low-dimensional factors arefound for the sake of dimensionality reduction. For their stochastic andgenerative behaviour, these models can also be used for generation of new datapoints in the data space. In this paper, we first start with variationalinference where we derive the Evidence Lower Bound (ELBO) and ExpectationMaximization (EM) for learning the parameters. Then, we introduce factoranalysis, derive its joint and marginal distributions, and work out its EMsteps. Probabilistic PCA is then explained, as a special case of factoranalysis, and its closed-form solutions are derived. Finally, VAE is explainedwhere the encoder, decoder and sampling from the latent space are introduced.Training VAE using both EM and backpropagation are explained.

Factor Analysis, Probabilistic Principal Component Analysis, Variational Inference, and Variational Autoencoder: Tutorial and Survey