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I have the following problem. Given a set of correlated binary variables, can I determine the joint probability from the correlation function?

{X_{i}} is a set of binary variables

Pr(X_{i}=1) = p and Pr(X_{i}=0) = q for all i

Corr(X_{i}X_{j}) = c_{ij}

c_{ij}is symmetric

Now how can I determine the joint probability Pr({X_{i}, X_{j}, X_{k}....})

For the joint probability of two variables I think I have the answer.

Noting that c_{ij}= (E(X_{i}X_{j}) - p^{2}) / pq, and using the notation {X_{i}=x_{i},X_{j}=x_{i}} -> {x_{i},x_{j}}

I have

Pr( {1,1} )= E(X_{i}X_{j}) = p*q*c_{ij}+ p^{2}

and by symmetry

Pr( {0,0} ) = p*q*c_{ij}+ q^{2}

and Pr( {0,1} ) = Pr( {1,0} ) = ( 1-Pr({1,1})-Pr({0,0}) ) / 2 = p*q*(1- c_{ij})

using that p+q = 1

How can I proceed to get Pr( {X_{i},X_{j}, X_{k}} ), and generally Pr( {X_{i},X_{j}, X_{k}, ….} )? I'm I missing something obvious? Any help or input is highly appreciated.

best

t

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# Joint Probability From Correlation Function

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