Re: Bayesian estimation of structured correlation/covariance

Ben Lee wrote:
>> Of course, I can measure several paths, so from what I've read, I
> should formulate this as covariance matrix estimation with the
Wishart
> distribution.
>
> I'd prefer correlation estimation which I understand from a
bivariate
> angle with the Fisher transform - but from what I've read from
> discussion in this group, the multivariate case gets quite nasty.
>

For your situation you might need to avoid something which make the
large step of using the Wishart/ inverse Wishart distribution for the
covariance matrix. This is because of the structured nature of the
correlation/covariance you want to impose. An alternative is to become
invested (more fully invested) in the modern computational approaches
to Bayesian statistics based on MCMC (Monte Carlo Markov Chains).

For example, in one approach based on "Gibbs sampling" you can arrive
a situation where your compuations for the covariance matrix are
converted to a case where the elements of the sample covariance matrix
can be treated as regression coefficents in a simplified problem that
allows their joint distribution to be described as multivariate
Normal. (The simplified problem arises because certain of the model
parameters are being temporarily treated as known). Your constraints
are then converted into information about the mean values for certain
of the marginal distributions or for certain linear combinations of
the marginal variables (the regression coefficients). The known
information is then implemented in the computations essentially by
using the theory for going from a joint multivariate Normal
distribution for (X,Y) to the conditional distribution (X|Y), where
here for example Y might be the set of sample regression coefficients
for pairs of variables where the true covariance (and regression
coefficent) is known to be zero. This works for the "known zero
correlation case" and possibly for the "known equal correlation case",
but you would need to look into rthe theory in more detail.

The above is very much a from-first-principles approach that would
need substantial effort to work out in detail. You may find that
literature already exists for the general problem of structured
correlation matrices, particularly for cases where some correlations
are assumed to be zero.

If you don't find an MCMC approach appealing you may still be best to
switch from viewing your problem as one of estimating the covarince
matrix, to one where you are estimating a set of regression
relationships.

David Jones


.

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