Variational principal components

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Abstract
One of the central issues in the use of princi- pal component analysis (PCA) for data mod- elling is that of choosing the appropriate num- ber of retained components. This problem was recently addressed through the formulation of a Bayesian treatment of PCA (Bishop, 1999a) in terms of a probabilistic latent variable model. A central feature of this approach is that the effec- tive dimensionality of the latent space (equiv- alent to the number of retained principal com- ponents) is determined automatically as part of the Bayesian inference procedure. In com- mon with most non-trivial Bayesian models, however, the required marginalizations are an- alytically intractable, and so an approximation scheme based on a local Gaussian representa- tion of the posterior distribution was employed. In this paper we develop an alternative, varia- tional formulation of Bayesian PCA, based on a factorial representation of the posterior distri- bution. This approach is computationally effi- cient, and unlike other approximation schemes, it maximizes a rigorous lower bound on the marginal log probability of the observed data.

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