  The projection predictive variable selection is a decision-theoretically
justified Bayesian variable selection approach achieving an outstanding
trade-off between predictive performance and sparsity. Its projection problem
is not easy to solve in general because it is based on the Kullback-Leibler
divergence from a restricted posterior predictive distribution of the so-called
reference model to the parameter-conditional predictive distribution of a
candidate model. Previous work showed how this projection problem can be solved
for response families employed in generalized linear models and how an
approximate latent-space approach can be used for many other response families.
Here, we present an exact projection method for all response families with
discrete and finite support, called the augmented-data projection. A simulation
study for an ordinal response family shows that the proposed method performs
better than or similarly to the previously proposed approximate latent-space
projection. The cost of the slightly better performance of the augmented-data
projection is a substantial increase in runtime. Thus, in such cases, we
recommend the latent projection in the early phase of a model-building workflow
and the augmented-data projection for final results. The ordinal response
family from our simulation study is supported by both projection methods, but
we also include a real-world cancer subtyping example with a nominal response
family, a case that is not supported by the latent projection.
