Quantify the approximation gap for asymmetric decision costs

Quantify the gap between the single Bayes-factor threshold with a sign check and the optimal policy under a continuous symmetric normal prior when false-positive and false-negative costs are asymmetric.

Background

The paper extends its Bayes-factor threshold analysis from a two-point hypothesis model to a continuous prior, specifically a symmetric normal prior for the treatment effect. Under asymmetric false-positive and false-negative costs, the optimal continuation boundaries in posterior-mean space are themselves asymmetric, whereas a single Bayes-factor threshold based on the symmetric prior depends only on the absolute posterior mean and therefore imposes a symmetry constraint.

The authors expect this constrained Bayes-factor rule to be close to optimal for moderate cost asymmetry, but they do not quantify the size of the gap. Determining this gap would clarify when the practically simpler directional Bayes-factor rule is an adequate approximation to the fully optimal sequential decision policy.

References

For asymmetric costs ($K_I \neq K_{II}$), the optimal thresholds satisfy $m_{\mathrm{high} \neq -m_{\mathrm{low}}$ and the single BF threshold is a constrained approximation to the optimal policy. Since the symmetric constraint is not binding at $K_I = K_{II}$, we expect the gap to be small for moderate asymmetry, though we have not quantified it.

— Bayesian Inference Procedures for A/B Testing: An Overview  (2608.12949 - Schultzberg et al., 13 Aug 2026) in Appendix, Section “BF Threshold Structure for General Cost Functions,” subsection “Extension to continuous priors”