  One of the main concepts in quantum physics is a density matrix, which is a
symmetric positive definite matrix of trace one. Finite probability
distributions are a special case where the density matrix is restricted to be
diagonal. Density matrices are mixtures of dyads, where a dyad has the form uu'
for any any unit column vector u. These unit vectors are the elementary events
of the generalized probability space. Perhaps the simplest case to see that
something unusual is going on is the case of uniform density matrix, i.e. 1/n
times identity. This matrix assigns probability 1/n to every unit vector, but
of course there are infinitely many of them. The new normalization rule thus
says that sum of probabilities over any orthonormal basis of directions is one.
We develop a probability calculus based on these more general distributions
that includes definitions of joints, conditionals and formulas that relate
these, i.e. analogs of the theorem of total probability, various Bayes rules
for the calculation of posterior density matrices, etc. The resulting calculus
parallels the familiar 'classical' probability calculus and always retains the
latter as a special case when all matrices are diagonal.
  Whereas the classical Bayesian methods maintain uncertainty about which model
is 'best', the generalization maintains uncertainty about which unit direction
has the largest variance. Surprisingly the bounds also generalize: as in the
classical setting we bound the negative log likelihood of the data by the
negative log likelihood of the MAP estimator.
