Bayesian Inference on the Stiefel Manifold: Models, Applications and Algorithms

In finance, it is crucial to use recent data to model the relationship between the companies since the market environment is evolving constantly. In particular, estimating time-varying covariance matrices has been an important topic for both portfolio optimization and risk management. Market measures such as betas for companies, beta dispersion, and market volatility are also closely related to the eigenvectors and eigenvalues of the covariance matrices. The current approaches for dynamic covariance estimation are focused on vector autoregressive processes and have shared parameters for the eigenvalues and eigenvectors. This inevitably introduces dependencies and fails to reveal the relationships between the model parameters. We contribute to the field of time- varying covariance estimation by proposing a Bayesian autoregressive model on the Stiefel manifold for high dimensional data. Our model considers the eigenvalues and eigenvectors separately, and provides a reliable solution to the relationships between the eigenvalues and eigenvectors. To our knowledge, this is the first attempt for an autoregressive time series model on the Stiefel manifold, and it can be extended to a class of models that are widely applicable to datasets in finance, biology, climate changes, etc.Our Bayesian model involves sampling and inference on the Stiefel manifold, which has been a challenging task. We contribute to Bayesian modeling on the Stiefel manifold by writing a new package using the Stan program. In our StanStiefel package, we extend the sampling method in [1], and propose novel parameter inference methods for popular distributions. Our package takes much less time to generate comparable amount of effective samples than the rstiefel package, especially in high dimensions