Decomposition of expectation values

Question

I have four (Bernoulli distributed) random variables, $S_1$, $P_1$, $S_2$, $P_2$, with the following properties:

• $S_1$ and $P_2$ are independent
• $S_2$ and $P_1$ are independent

all other combinations have correlations.

If I want to evaluate the expectation value $$ E[P_1 P_2 S_1 S_2] $$ is there any way to decompose it as correlators e.g. $$ E[P_1 P_2], E[S_1 S_2], ... $$

Can one use some kind of "conditioned" expectation values?

EDIT: each random variable can take two values e.g. +1, 0.

Answer 1

Just expand the joint probability as usual $$ \eqalign{ E[P_1,P_2,S_1,S_2]&=P(P_1,P_2,S_1,S_2)\\ &=P(P_1)P(P_2,S_1,S_2 | P_1)\\ &=P(P_1)P(P_2|P_1)P(S_1,S_2 | P_1,P_2)\\ &=P(P_1)P(P_2|P_1)P(S_1|P_1,P_2)P(S_2 | P_1,P_2,S_1)\\ &=P(P_1)P(P_2|P_1)P(S_1|P_1)P(S_2 | P_2,S_1) } $$ the last line removes the conditions that are not necessary to specify given the independence conditions. You can rewrite the probabilities in that line as conditioned expectation values if you wish.