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Majorizing estimators and the approximation of #Pcomplete problems

Summary: Majorizing estimators and
the approximation of #P­complete problems
Leonard J. Schulman \Lambda Vijay V. Vazirani \Lambda
A key step in counting via sampling is constructing an un­
biased estimator, X, for the parameter ` in question, and
proving a bound on its second moment, E(X 2 ). A key ap­
plication of this method is to obtaining a FPRAS for a #P­
complete problem; a FPRAS results if the ratio r = E(X 2 ) 1=2
is polynomially bounded in the size of the input. We show
that if no additional information is available about the dis­
tribution of X, then this condition is also necessary.
The proof involves establishing a new optimality result in
parametric statistics. We introduce the notion of a majoriz­
ing estimator, a very strict optimality requirement that we
need for making worst­case (over inputs) and in­probability
(of falling in the desired accuracy range of the parameter `)
statements. We show that for the problem of estimating the
mean of a Gaussian distribution (from the variable­location,


Source: Abu-Mostafa, Yaser S. - Department of Mechanical Engineering & Computer Science Department, California Institute of Technology


Collections: Computer Technologies and Information Sciences