  We formulate incomplete classical statistics for situations where the
knowledge about the probability distribution outside a local region is limited.
The information needed to compute expectation values of local observables can
be collected in a quantum mechanical state vector, whereas further statistical
information about the probability distribution outside the local region becomes
irrelevant. The translation of the available information between neighboring
local regions is expressed by a Hamilton operator. A quantum mechanical
operator can be associated to each local observable, such that expectation
values of ``classical'' observables can be computed by the usual quantum
mechanical rules. The requirement that correlation functions should respect
equivalence relations for local obeservables induces a non-commutative product
in classical statistics, in complete correspondence to the quantum mechanical
operator product. We also discuss the issue of interference and the complex
structure of quantum mechanics within our classical statistical setting.
