  Electromagnetic radiation plays a crucial role in various physical and
chemical processes. Hence, almost all astrophysical simulations require some
form of radiative transfer model. Despite many innovations in radiative
transfer algorithms and their implementation, realistic radiative transfer
models remain very computationally expensive, such that one often has to resort
to approximate descriptions. The complexity of these models makes it difficult
to assess the validity of any approximation and to quantify uncertainties on
the model results. This impedes scientific rigour, in particular, when
comparing models to observations, or when using their results as input for
other models. We present a probabilistic numerical approach to address these
issues by treating radiative transfer as a Bayesian linear regression problem.
This allows us to model uncertainties on the input and output of the model with
the variances of the associated probability distributions. Furthermore, this
approach naturally allows us to create reduced-order radiative transfer models
with a quantifiable accuracy. These are approximate solutions to exact
radiative transfer models, in contrast to the exact solutions to approximate
models that are often used. As a first demonstration, we derive a probabilistic
version of the method of characteristics, a commonly-used technique to solve
radiative transfer problems.
