[ADMB Users] citation for empirical Bayes variance computation

dave fournier otter at otter-rsch.com
Fri May 7 08:05:47 PDT 2010

It came out of my head.

Reasoning is as follows. Let the RE's be u and the other parameters
be x.

Let uhat(x) be the value of u which maximizes the function
(joint probability dist if you are a Bayesian) l(x,u)
of x and u for a given value of x. Then the delta method gives the
estimate for the variance of uhat as

           trans(uhat'(x)) * inv(log(L)_xx)* uhat'(x)

where L(x) = int l(x,u) du

If uhat(x) is known then  a candidate for the variance of u would be


so add them together to reflect the fact that uncertainly in the
x gives uncertainty in the value of uhat.
So it just seems like a reasonable calculation. Of course in nonlinear
models this approximation can be quite bad in more extreme cases.

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