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Torkel
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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.
best
t
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.
best
t
Last edited: