  Physics is based on probabilities as fundamental entities of a mathematical
description. Expectation values of observables are computed according to the
classical statistical rule. The overall probability distribution for one world
covers all times. The quantum formalism arises once one focuses on the
transport of the time-local probabilistic information from one hypersurface to
a neighboring one. Wave functions or the density matrix allow the formulation
of a general linear evolution law for classical statistics. The density matrix
for classical statistics is a powerful tool which allows us to implement for
generalized Ising models concepts as basis transformations, the momentum
observable and the associated Fourier representation, or the definition of
subsystems by subtraces of the density matrix. The association of operators to
observables permits the computation of expectation values in terms of the
density matrix by the usual quantum rule. We show that probabilistic cellular
automata are quantum systems in a formulation with discrete time steps and real
wave functions. The evolution operator for automata can be expressed in terms
of fermionic creation and annihilation operators. The time-local probabilistic
information amounts to a subsystem of the overall probabilistic system which is
correlated with its environment consisting of the past and future. Such
subsystems typically involve probabilistic observables for which only a
probability distribution for their possible measurement values is available.
Incomplete statistics does not permit to compute classical correlation
functions for arbitrary subsystem-observables since different overall
observables are mapped to the same subsystem observable. Bell's inequalities
are not generally applicable.
