  The aim of this paper is to show a connection between an extended theory of
statistical experiments on the one hand and the foundation of quantum theory on
the other hand. The main aspects of this extension are: One assumes a
hyperparameter space $\Phi$ common to several potential experiments, and a
basic symmetry group G associated with this space. The parameter \theta_{a} of
a single experiment, looked upon as a parametric function $\theta_{a}(\cdot)$
on $\Phi$, is said to be permissible if G induces in a natural way a new group
on the image space of the function. If this is not the case, it is arranged for
by changing from G to a subgroup $G_{a}$. The Haar measure of this subgroup
(confined to the spectrum; see below) is the prefered prior when the parameter
is unknown. It is assumed that the hyperparameter itself can never be
estimated, only a set of parametric functions. Model reduction is made by
restricting the space of complex `wave' functions, also regarded as functions
on $\Phi$, to an irreducible invariant subspace $\mathcal{M}$ under G. The
spectrum of a parametric function is defined from natural group-theoretical and
statistical considerations. We establish that a unique operator can be
associated with every parametric functions $\theta_{a}(\cdot)$, and essentially
all of the ordinary quantum theory formalism can be retrieved from these and a
few related assumptions. In particular the concept of spectrum is consistent.
The scope of the theory is illustrated on the one hand by the example of a spin
1/2 particle and a related EPR discussion, on the other hand by a simple
macroscopic example.
