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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.