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I have a question about a joint distribution calculated in a paper I am reading.

There are three random variable a, b and c such that $$ a,b,c \in \{+1,-1\} $$ and then the joint distribution is given by: $$ p(a,b,c) = \frac{1}{8}(1 + aE_{A} + bE_{B} + cE_{C} + abE_{AB} + acE_{AC} + bcE_{BC} + abcE_{ABC})$$ where $E_{A}$, $E_{B}$ and $E_{C}$ are the single-party marginals, $E_{AB}$, $E_{BC}$ and $E_{AC}$ the two-party marginals, and $E_{ABC}$ is the three-body correlator.

I feel like this should be something I should have seen in an introductory course on probability but I can't seem to prove it. Also if I convince myself that it's just adding up all the possible cases, I think 1 should be part of the other expected values (i.e. it is already considered in them) so I don't see the point in adding it separately. Also would we have the same expression if the possible values were $\{+1,0\}$ (or any other set of size 2) ?

I am used to the notation $E$ as the expected value, but this is totally unrelated to that and it is about marginals. Am I correct?

I would be pleased to see a complete proof or a link to study this fact.

It also adds:

Note that the positivity of $p(a,b,c)$ implies constraints on marginals, in particular $p(+ + +) + p(−−−) ≥ 0$ implies $$ E_{AB} + E_{AC} + E_{BC} ≥ −1.$$

which I don't understand.

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  • $\begingroup$ Can you add a link to the paper? $\endgroup$
    – Leander Tilsted Kristensen
    Nov 19, 2021 at 12:07
  • $\begingroup$ I think I heard the term 'Rademacher chaos' but not sure how deeply it is related to your question. Anyway, the expansion is more or less the result of the fact that monomials $\prod_{x\in A}x$ for $A\subseteq \{a,b,c\}$ spans the space of functions of the form $\{-1,1\}^3\to\mathbb{R}$. $\endgroup$
    – Sangchul Lee
    Nov 19, 2021 at 12:10
  • $\begingroup$ arxiv.org/abs/1906.06495 $\endgroup$
    – Pegi
    Nov 19, 2021 at 12:37
  • $\begingroup$ It is on the second page. $\endgroup$
    – Pegi
    Nov 19, 2021 at 12:38
  • $\begingroup$ The equations $(3)$-$(10)$ in the paper indicate that $E_A,E_B,E_C,E_{AB},E_{AC},E_{BC}$ and $E_{ABC}$ does indeed represent the expected values $E[A],E[B],E[C],E[AB],E[AC],E[BC]$ and $E[ABC]$. $\endgroup$
    – Leander Tilsted Kristensen
    Nov 19, 2021 at 14:27

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