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Joint Probability From Correlation Function

  1. Mar 23, 2010 #1
    Dear all

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

    {Xi} is a set of binary variables
    Pr(Xi=1) = p and Pr(Xi=0) = q for all i
    Corr(Xi Xj) = cij

    cij is symmetric

    Now how can I determine the joint probability Pr({Xi, Xj, Xk ....})
    For the joint probability of two variables I think I have the answer.
    Noting that cij= (E(Xi Xj) - p2) / pq, and using the notation {Xi=xi,Xj=xi} -> {xi,xj}

    I have
    Pr( {1,1} )= E(Xi Xj) = p*q*cij + p2

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

    and Pr( {0,1} ) = Pr( {1,0} ) = ( 1-Pr({1,1})-Pr({0,0}) ) / 2 = p*q*(1- cij )
    using that p+q = 1

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

    Last edited: Mar 24, 2010
  2. jcsd
  3. Nov 23, 2010 #2
    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.

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