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Passing Covariance Matrix in Likelihood Maximization

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Marco Lago
Marco Lago on 27 Feb 2021
Answered: Jeff Miller on 27 Feb 2021
I've a question about code design for a numerical optimization of the likelihood function of the sort:
LL = LogLik(x,y,Param,L,Ps,P)
My objective function makes use of the multivariate normal probability density in the following fashion:
eta = mvnpdf(y,x*Phi,Sigma);
Where sigma is passed throug the parameter vector 'Param' (together with Phi). I then try to numerically optimize that through fmincon but then Sigma shoud be a positive-definite covariance matrix, condition that is not always defined and causes an error.
Results = fmincon(@(Param)LogLik(X,Y1,Param,L,Ps,P),x0,[],[],Aeq',beq,[],[],[],options)
I'm asking your help as I suspect this is not the right approach to solve the problem at hand.

Accepted Answer

Jeff Miller
Jeff Miller on 27 Feb 2021
Sometimes this kind of problem can be solved by adding code within the LogLik function to make sure that a legal (i.e., positive definite) value of Sigma is computed from any possible combination of Param values. So, the Param values are not themselves elements of Sigma, but rather values that determine the elements of Sigma. As an example with Sigma determined by parameters 2-4:
function LL = LogLik(...
VarX = Param(2)^2; % Make sure positive
VarY = Param(3)^2; % Make sure positive
rhoXY = Param(4) / (abs(Param(4)+1)); % Make sure correlation between +/- 1
covarXY = rhoXY*sqrt(VarX)*sqrt(VarY);
Sigma = [VarX covarXY;
covarXY VarY]
eta = mvnpdf(y,x*Phi,Sigma);
...
Hope that helps

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