Projection predictive variable selection for discrete response families with finite support

TL;DR

This paper introduces an exact projection method for Bayesian variable selection in models with discrete, finite support responses, improving accuracy at the cost of increased computation, and compares it to approximate methods.

Contribution

It presents the augmented-data projection, an exact solution for all discrete finite support response families, extending previous approximate latent-space approaches.

Findings

The augmented-data projection outperforms or matches the approximate method in simulations.

It is more computationally intensive, suitable for final model selection.

The method applies to both ordinal and nominal response families.

Abstract

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.