  A perennial objection against Bayes factor point-null hypothesis tests is
that the point-null hypothesis is known to be false from the outset. Following
Morey and Rouder (2011) we examine the consequences of approximating the sharp
point-null hypothesis by a hazy `peri-null' hypothesis instantiated as a narrow
prior distribution centered on the point of interest. The peri-null Bayes
factor then equals the point-null Bayes factor multiplied by a correction term
which is itself a Bayes factor. For moderate sample sizes, the correction term
is relatively inconsequential; however, for large sample sizes the correction
term becomes influential and causes the peri-null Bayes factor to be
inconsistent and approach a limit that depends on the ratio of prior ordinates
evaluated at the maximum likelihood estimate. We characterize the asymptotic
behavior of the peri-null Bayes factor and discuss how to construct peri-null
Bayes factor hypothesis tests that are also consistent.
