  Orthogonal polynomial approximations form the foundation to a set of
well-established methods for uncertainty quantification known as polynomial
chaos. These approximations deliver models for emulating physical systems in a
variety of computational engineering applications. In this paper, we describe a
Bayesian formulation of polynomial approximations capable of incorporating
uncertainties in input data. Through different priors in a hierarchical
structure, this permits us to incorporate expert knowledge on the inference
task via different approaches. These include beliefs of sparsity in the model;
approximate knowledge of the polynomial coefficients (e.g. through low-fidelity
estimates) or output mean, and correlated models that share similar functional
and/or physical behaviours. We show that through a Bayesian framework, such
prior knowledge can be leveraged to produce orthogonal polynomial
approximations with enhanced predictive accuracy.
