Actually it is not that difficult to parameterize the covariance matrix
so that the
optimization is unconstrained. first parameterize the correlation matrix
and the
standard deviations separately. the std devs can be parameterized as
sigma_i=exp(x_i) 1<=i<=n
For the correlation matrix parameterize it in terms of its choleski
decomposition.
this is a lower triangular matrix
1 0 0
a_1 a_2 0
b _1 b_2 b_3
....
such that the norm of each row is 1
to ensure this form start with
1
y_1 1
y_2 y_3 1
...
and normalize each row to have norm 1. There are n*(n-1)/2 of these y's.
together with the n x_i's you have n*(n+1)/2 parameters as you should.
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