  Statistical modeling can involve a tension between assumptions and
statistical identification. The law of the observable data may not uniquely
determine the value of a target parameter without invoking a key assumption,
and, while plausible, this assumption may not be obviously true in the
scientific context at hand. Moreover, there are many instances of key
assumptions which are untestable, hence we cannot rely on the data to resolve
the question of whether the target is legitimately identified. Working in the
Bayesian paradigm, we consider the grey zone of situations where a key
assumption, in the form of a parameter space restriction, is scientifically
reasonable but not incontrovertible for the problem being tackled.
Specifically, we investigate statistical properties that ensue if we structure
a prior distribution to assert that `maybe' or `perhaps' the assumption holds.
Technically this simply devolves to using a mixture prior distribution putting
just some prior weight on the assumption, or one of several assumptions,
holding. However, while the construct is straightforward, there is very little
literature discussing situations where Bayesian model averaging is employed
across a mix of fully identified and partially identified models.
