  There are many issues that can cause problems when attempting to infer model
parameters from data. Data and models are both imperfect, and as such there are
multiple scenarios in which standard methods of inference will lead to
misleading conclusions; corrupted data, models which are only representative of
subsets of the data, or multiple regions in which the model is best fit using
different parameters. Methods exist for the exclusion of some anomalous types
of data, but in practice, data cleaning is often undertaken by hand before
attempting to fit models to data. In this work, we will employ hierarchical
Bayesian data selection; the simultaneous inference of both model parameters,
and parameters which represent our belief that each observation within the data
should be included in the inference. The aim, within a Bayesian setting, is to
find the regions of observation space for which the model can well-represent
the data, and to find the corresponding model parameters for those regions. A
number of approaches will be explored, and applied to test problems in linear
regression, and to the problem of fitting an ODE model, approximated by a
finite difference method. The approaches are simple to implement, can aid
mixing of Markov chains designed to sample from the arising densities, and are
broadly applicable to many inferential problems.
