Discussion of “Optimal Test Procedures for Multiple Hypotheses Controlling the Familywise Expected Loss” by Willi Maurer, Frank Bretz, and Xiaolei Xun

Rejoinder to Discussions on “Optimal Test Procedures for Multiple Hypotheses Controlling the Familywise Expected Loss”

“…LaVange et al ( 2023) discussed a simulation-based framework for Bayesian designs relying on recent developments by Alt et al (2022) for FWER control. Banerjee (2023) focused mainly on Bayesian modeling for complex spatial data and followed the approach by Müller et al (2004) for FDR control. It is worth pointing out a result from Berger (1985, p. 159), which shows that minimizing the Bayes risk 𝐸 𝜋(𝜃) {𝐸 𝑋 𝜃 [𝐿(𝑑(𝑋)), 𝜃]} with a prior distribution 𝜋(𝜃) and minimizing the posterior expected loss 𝐸 𝜋(𝜃|𝑥) [𝐿(𝑑(𝑥), 𝜃)] with a posterior distribution 𝜋(𝜃|𝑥) are equivalent problems.…”

“…We acknowledge that using a Bayesian approach could be attractive in certain applications. For example, in the spatial difference boundary detection problem discussed by Banerjee (2023), inference on spatial random effects is easier when using a Bayesian model than a frequentist model, as the author pointed out. Another example is the multiple decision problem introduced by Banerjee (2023) when investigating the set of hypotheses concerning all possible equal and unequal relationships among the effects.…”