  Hypothesis testing is an essential statistical method in psychology and the
cognitive sciences. The problems of traditional null hypothesis significance
testing (NHST) have been discussed widely, and among the proposed solutions to
the replication problems caused by the inappropriate use of significance tests
and $p$-values is a shift towards Bayesian data analysis. However, Bayesian
hypothesis testing is concerned with various posterior indices for significance
and the size of an effect. This complicates Bayesian hypothesis testing in
practice, as the availability of multiple Bayesian alternatives to the
traditional $p$-value causes confusion which one to select and why. In this
paper, we compare various Bayesian posterior indices which have been proposed
in the literature and discuss their benefits and limitations. Our comparison
shows that conceptually not all proposed Bayesian alternatives to NHST and
$p$-values are beneficial, and the usefulness of some indices strongly depends
on the study design and research goal. However, our comparison also reveals
that there exist at least two candidates among the available Bayesian posterior
indices which have appealing theoretical properties and are, to our best
knowledge, widely underused among psychologists.
