Joint Probability From Correlation Function

[Torkel]

Dear all

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) - p²) / 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 + p²

and by symmetry
Pr( {0,0} ) = p*q*c_ij + q²

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

 

[omrit]

You will need higher moments - the pairwise correlations are not enough for n>2. Look into Teugels (1990) ‘Some Representations of the Multivariate Bernoulli and Binomial Distributions’, where he provides a formulation for a general multivariate Bernoulli with dependencies, for n dimensions.

Omri.