We examine the conditions under which descriptive inference can be based
directly on the observed distribution in a non-probability sample, under both
the super-population and quasi-randomisation modelling approaches. Review of
existing estimation methods reveals that the traditional formulation of these
conditions may be inadequate due to potential issues of under-coverage or
heterogeneous mean beyond the assumed model. We formulate unifying conditions
that are applicable to both type of modelling approaches. The difficulties of
empirically validating the required conditions are discussed, as well as valid
inference approaches using supplementary probability sampling. The key message
is that probability sampling may still be necessary in some situations, in
order to ensure the validity of descriptive inference, but it can be much less
resource-demanding provided the presence of a big non-probability sample
