[ADMB Users] Improving importance sampling for mixed models

[dave fournier]

There was a typically bad idea expressed on the R list today.

https://stat.ethz.ch/pipermail/r-sig-mixed-models/2013q1/020017.html

Ir reminded me of something I was thinking about a while back to improve the
performance of importing sampling for mixed models.  The idea is to
create negatively correlated  random points.  It is based on the idea
that for a std multivariate normal vector, the norm has a probability
density function

         c * r^{n-1} exp(-r^2/2)

for a constant c.
In general this produces a CDF based on the gamma function.
The idea is to generate a sample of multivariate normals and then so 
scale them
based on the inverse of the CDF.  We then restrict each one to the shell
based on its norm and maximize the distance between them all.
For 2 dimensions one can plot the results.  Here are 250 bivariate normals.
You can see that some of them are almost on top each other.  This produces
inefficient sampling.



Now here are the results of rescaling the norms


and finally what we get after maximizing the distance between them.



Of course one does not need to use the inverse of the CDF based on the 
normal.
a fatter tailed distribution could be used.

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