Spurious-inclusion control for nonparametric priors

Establish the spurious-inclusion control condition for nonparametric priors in Bayesian variable-selection models, thereby bounding posterior support on structures that include a variable but represent its effect incorrectly.

Background

The detection-threshold theory separates informative structures, which can represent how a variable affects the truth, from spurious structures, which include the variable but use it in an inappropriate way. The lower bounds for posterior detection rely on Assumption c6, a Bayes-factor bound controlling the posterior contribution of such spurious structures.

The paper explicitly identifies verification of this condition for nonparametric priors as unresolved. Establishing it would extend the sharp prior-weight detection theory beyond the settings where the required Bayes-factor behavior is currently assumed or verified.

References

On the theory side, the remaining open problems are spurious-inclusion control (assumption \Cref{ass:c6}) for nonparametric priors, an explicit second-order term in the detection threshold, and conditions under which training provably achieves uniform amortization.

— SelectPFN: Amortized Bayesian Variable Selection for Sparse Nonlinear Regression with Prior-Data Fitted Networks  (2610.21950 - Luo, 15 Oct 2026) in Section 4.5, Discussion, paragraph “Extensions”