  Statistical models typically capture uncertainties in our knowledge of the
corresponding real-world processes, however, it is less common for this
uncertainty specification to capture uncertainty surrounding the values of the
inputs to the model, which are often assumed known. We develop general
modelling methodology with uncertain inputs in the context of the Bayes linear
paradigm, which involves adjustment of second-order belief specifications over
all quantities of interest only, without the requirement for probabilistic
specifications. In particular, we propose an extension of commonly-employed
second-order modelling assumptions to the case of uncertain inputs, with
explicit implementation in the context of regression analysis, stochastic
process modelling, and statistical emulation. We apply the methodology to a
regression model for extracting aluminium by electrolysis, and emulation of the
motivating epidemiological simulator chain to model the impact of an airborne
infectious disease.
