Why do we draw parameters to draw from a marginal distribution that does not contain the parameters?

A revealing question on X validated of a simulation concept students (and others) have trouble gripping with. Namely using auxiliary variates to simulate from a marginal distribution, since these auxiliary variables are later dismissed and hence appear to them (students) of no use at all. Even after being exposed to the accept-reject algorithm. Or to multiple importance sampling. In the sense that a realisation of a random variable can be associated with a whole series of densities in an importance weight, all of them being valid (but some more equal than others!).

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