General Bayesian Inference over the Stiefel Manifold via the Givens Representation

We introduce an approach based on the Givens representation that allows for a routine, reliable, and flexible way to infer Bayesian models with orthogonal matrix parameters. This class of models most notably includes models from multivariate statistics such factor models and probabilistic principal component analysis (PPCA). Our approach overcomes several of the practical barriers to using the Givens representation in a general Bayesian inference framework. In particular, we show how to inexpensively compute the change-of-measure term necessary for transformations of random variables. We also show how to overcome specific topological pathologies that arise when representing circular random variables in an unconstrained space. In addition, we discuss how the alternative parameterization can be used to define new distributions over orthogonal matrices as well as to constrain parameter space to eliminate superfluous posterior modes in models such as PPCA. While previous inference approaches to this problem involved specialized updates to the orthogonal matrix parameters, our approach lets us represent these constrained parameters in an unconstrained form. Unlike previous approaches, this allows for the inference of models with orthogonal matrix parameters using any modern inference algorithm including those available in modern Bayesian modeling frameworks such as Stan, Edward, or PyMC3. We illustrate with examples how our approach can be used in practice in Stan to infer models with orthogonal matrix parameters, and we compare to existing methods.