  Dimension reduction is useful for exploratory data analysis. In many
applications, it is of interest to discover variation that is enriched in a
"foreground" dataset relative to a "background" dataset. Recently, contrastive
principal component analysis (CPCA) was proposed for this setting. However, the
lack of a formal probabilistic model makes it difficult to reason about CPCA
and to tune its hyperparameter. In this work, we propose probabilistic
contrastive principal component analysis (PCPCA), a model-based alternative to
CPCA. We discuss how to set the hyperparameter in theory and in practice, and
we show several of PCPCA's advantages, including greater interpretability,
uncertainty quantification, robustness to noise and missing data, and the
ability to generate data from the model. We demonstrate PCPCA's performance
through a series of simulations and experiments with datasets of gene
expression, protein expression, and images.
