🚀 go-pugleaf

I obtain parameter estimates via maximum likelihood where
my model is in the standard reduced form y = f(p), y are the
data and p are the parameters. I assume that the distribution
of the model + measurement errors is normal with zero mean
and known covariance matrix Ve. Thus i am solving the optimization
problem:

	min Tr(y - f(p))Inv(Ve)(y - f(p))
         p

where Tr() is the transpose and Inv() is the inverse. I have a question
about the mathematics of deriving the standard linear estimate of the
covariance matrix of the sampling distribution of the parameter estimates, p*.

To form my question, let me go briefly into my interpretation of the
covariance matrix derivation. First of all, the optimal value function of
the max. like. problem depends on data, that is p* = p*(y). The definition of
the covariance matrix of the sampling distribution, Vp, is:

	Vp = E[(dp*)Tr(dp*)]

where the perturbation of the optimal parameters, dp* is estimated by the
linear terms of a Taylor series:

	dp* ~ Tr(dp*/dy)dy

where dp*/dy are the partial derivatives of the function p*(y) and are 
evaluated at (y,p*). Substituting this into the definition of Vp gives:

	Vp ~ Tr(dp*/dy)E[(dy)Tr(dy)](dp*/dy)
	Vp ~ Tr(dp*/dy)Vy(dp*/dy)


Now the rest of the derivation continues with the evaluation of the partial
derivatives (dp*/dy), often requiring additional approximations based on
small residuals and other assumptions. The true dp*/dy would certainly 
require second partials of the log likelihood function, whereas certain
approximations (the most common, i believe) use only first partials. In any
case, the partials dp*/dy will certainly involve the combined error covariance
matrix Ve.

My question can now (finally) be stated: Is the covariance matrix Vy above
one and the same with Ve, the covariance of the model + measurement errors?
I have been really confused by this for a long time. Just intuitively,
but perhaps naively, i'd have thought that Vy only includes the *measurement*
errors, and not the model errors. All of the derivations i've seen, however,
seem to at some stage treat both Vy and Ve as the same. For example, one
standard final results is the following:

	Vp ~ Inv[Tr(df/dp)Inv(V)(df/dp)]

But, i must confess, i really am unsure how to interpret the covariance
V in the above equation.

Thank you for your time and assistance in helping me with this problem.    

jim uber
dept. of civil & env. engineering
univ. of cincinnati
juber@uceng.uc.edu

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james uber
juber@uceng.uc.edu