  A perennial objection against Bayes factor point-null hypothesis tests is
that the point-null hypothesis is known to be false from the outset. 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 briefly discuss suggestions on how to construct peri-null Bayes factor
hypothesis tests that are also consistent.
