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

Parameter Restrictions for the Sake of Identification: Is there Utility in Asserting that Perhaps a Restriction Holds?