  We introduce a formulation of quantum theory (QT) as a general probabilistic
theory but expressed via quasi-expectation operators (QEOs). This formulation
provides a direct interpretation of density matrices as quasi-moment matrices.
Using QEOs, we will provide a series of representation theorems, a' la de
Finetti, relating a classical probability mass function (satisfying certain
symmetries) to a quasi-expectation operator. We will show that QT for both
distinguishable and indistinguishable particles can be formulated in this way.
Although particles indistinguishability is considered a truly "weird" quantum
phenomenon, it is not special. We will show that finitely exchangeable
probabilities for a classical dice are as weird as QT. Using this connection,
we will rederive the first and second quantisation in QT for bosons through the
classical statistical concept of exchangeable random variables. Using this
approach, we will show how classical reality emerges in QT as the number of
identical bosons increases (similar to what happens for finitely exchangeable
sequences of rolls of a classical dice).
