## AbstractLinear data analysis methods such as factor analysis (FA), independent component analysis (ICA) and blind source separation (BSS) as well as state-space models such as the Kalman filter model are used in a wide range of applications. In many of these, linearity is just a convenient approximation while the underlying effect is nonlinear. It would therefore be more appropriate to use nonlinear methods. In this work, nonlinear generalisations of FA and ICA/BSS are presented. The methods are based on a generative model, with a multilayer perceptron (MLP) network to model the nonlinearity from the latent variables to the observations. The model is estimated using variational Bayesian learning. The variational Bayesian method is well-suited for the nonlinear data analysis problems. The approach is also theoretically interesting, as essentially the same method is used in several different fields and can be derived from several different starting points, including statistical physics, information theory, Bayesian statistics, and information geometry. These complementary views can provide benefits for interpretation of the operation of the learning method and its results. Much of the work presented in this thesis consists of improvements that make the nonlinear factor analysis and blind source separation methods faster and more stable, while being applicable to other learning problems as well. The improvements include methods to accelerate convergence of alternating optimisation algorithms such as the EM algorithm and an improved approximation of the moments of a nonlinear transform of a multivariate probability distribution. These improvements can be easily applied to other models besides FA and ICA/BSS, such as nonlinear state-space models. A specialised version of the nonlinear factor analysis method for post-nonlinear mixtures is presented as well.
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