Tutorial presented at the 1999 NIPS Conference by Zoubin Ghahramani and
Sam Roweis
Abstract: Many of the methods used for clustering, dimensionality
reduction, source separation, time series modeling, and other
classical problems in unsupervised data modeling are closely
related to each other. The focus of this tutorial is to present a
consistent unified picture of how these methods, which have been
developed and rediscovered in several different fields, are
variants of each other, and how a single framework can be used to
develop learning algorithms for all of them. We will start from a
humble Gaussian model, to describe how continuous state models such
as factor analysis, principal components analysis (PCA) and
independent components analysis (ICA) are related to each other. We
will then motivate discrete state mixture models and vector
quantization. Mixture models and factor analysis are then extended
to model time series data, and result in hidden Markov models
(HMMs) and linearGaussian dynamical systems (a.k.a. statespace
models), respectively.

All of these models can be described within the framework of probabilistic graphical models, which we will briefly introduce. In this framework it becomes easy to explore variants and hybrids (such as mixtures of factor analyzers and switching statespace models) which are potentially powerful tools. This framework also makes it clear that the same general probability propagation algorithm can be used to infer the hidden (i.e. latent) variables in all these models, and that the EM algorithm can be used to learn the maximum likelihood (ML) parameters. In the latter part of the tutorial we will focus on approximate inference techniques for models in which probability propagation is intractable, and on variational methods for Bayesian model averaging which can overcome the overfitting and model selection problems in ML learning. Matlab demos will be used to demonstrate some of the models and algorithms. 