  We derive essential elements of quantum mechanics from a parametric structure
extending that of traditional mathematical statistics. The basic setting is a
set $\mathcal{A}$ of incompatible experiments, and a transformation group $G$
on the cartesian product $\Pi$ of the parameter spaces of these experiments.
The set of possible parameters is constrained to lie in a subspace of $\Pi$, an
orbit or a set of orbits of $G$. Each possible model is then connected to a
parametric Hilbert space. The spaces of different experiments are linked
unitarily, thus defining a common Hilbert space $\mathbf{H}$. A state is
equivalent to a question together with an answer: the choice of an experiment
$a\in\mathcal{A}$ plus a value for the corresponding parameter. Finally,
probabilities are introduced through Born's formula, which is derived from a
recent version of Gleason's theorem. This then leads to the usual formalism of
elementary quantum mechanics in important special cases. The theory is
illustrated by the example of a quantum particle with spin.
