  Functional principal components analysis is a popular tool for inference on
functional data. Standard approaches rely on an eigendecomposition of a
smoothed covariance surface in order to extract the orthonormal functions
representing the major modes of variation. This approach can be a
computationally intensive procedure, especially in the presence of large
datasets with irregular observations. In this article, we develop a Bayesian
approach, which aims to determine the Karhunen-Lo\`eve decomposition directly
without the need to smooth and estimate a covariance surface. More
specifically, we develop a variational Bayesian algorithm via message passing
over a factor graph, which is more commonly referred to as variational message
passing. Message passing algorithms are a powerful tool for compartmentalizing
the algebra and coding required for inference in hierarchical statistical
models. Recently, there has been much focus on formulating variational
inference algorithms in the message passing framework because it removes the
need for rederiving approximate posterior density functions if there is a
change to the model. Instead, model changes are handled by changing specific
computational units, known as fragments, within the factor graph. We extend the
notion of variational message passing to functional principal components
analysis. Indeed, this is the first article to address a functional data model
via variational message passing. Our approach introduces two new fragments that
are necessary for Bayesian functional principal components analysis. We present
the computational details, a set of simulations for assessing accuracy and
speed and an application to United States temperature data.
