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Topic: Does AD work on function that involves Monte Carlo?
Replies: 5   Last Post: Sep 8, 2007 4:42 PM

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 Brian Borchers Posts: 148 Registered: 12/6/04
Re: Does AD work on function that involves Monte Carlo?
Posted: Sep 8, 2007 4:42 PM

hrubin@odds.stat.purdue.edu wrote:
>Get the random arguments (and use a good random number
>generator, which usually is not provided) before computing
>the derivatives. Use the SAME arguments for computing any
>function and all its derivatives.

This is a clever way to avoid the problem with independent errors in
the Monte Carlo estimates. It should work for simple Monte Carlo
integration over a fixed region of integration, where the random
numbers are only used to pick points at which to evaluate the
function. You certainly don't want to use the same set of
pseudorandom points each time if the region of integration depends on
the parameters so that it changes from one evaluation of the objective
function to the next.

With more complicated integration schemes this might be difficult to
make work. For example, your integration code might use the function
values to dynamically partition the region over which the function is
being integrated so that it can focus in on "important" areas. In
such cases you wouldn't be able to use the same set of pseudorandom
points for each integration.

A further issue is that you're now minimizing the approximation of the
original function obtained with this particular sequence of pseudo
random points. Although this approximation should be smooth, the
error in this approximation might vary from point to point in the
parameter space that you're optimizing over. You might well end up
finding a minimum of the approximation that is far from any minimum of
the original objective function.

The original poster asked for a pointer to information on response
surface methods. You can find an introduction to the topic in
the textbook, "Simulation Modeling and Analysis, 3rd ed." by Law
and Kelton. Myers and Montgomery have a book titled "Response Surface
Methodology: Process and Product Optimization Using Designed Experiments."

--
Brian Borchers borchers@nmt.edu
Department of Mathematics http://www.nmt.edu/~borchers/
New Mexico Tech Phone: 505-835-5813
Socorro, NM 87801 FAX: 505-835-5366

Date Subject Author
9/5/07 housing2006@gmail.com
9/6/07 Jim Turner
9/6/07 Brian Borchers
9/6/07 housing2006@gmail.com
9/8/07 Herman Rubin
9/8/07 Brian Borchers