694

**Double Importance Sampling**

**Valérie Ventura**

Revised 4/00

### Abstract:

Assume that we want to estimate
via simulation,
for , and for several functions ,where *X* is a random variable with density .The importance sampling identity can be used to write
which then can be estimated by
where is a random sample from *g*.
For importance sampling to be efficient though,
sampling from *g* should be easy,
*g* must provide adequate coverage
of the sample space of possibly many densities ,and ideally it must be chosen to minimize the variance
of the resulting estimates of .This is a lot to achieve,
particularly since the goals might be conflicting.
Moreover,
if several characteristics must be estimated,
the method is unavoidably limited as a variance reduction technique, because
*g* can only be optimal for one particular ; worse, it can potentially
be very non-optimal for other characteristics.
On the other hand, double importance sampling
allows to achieve all the goals: ease of sampling, and theoretically perfect estimation
of an arbitrarily large number of quantities .One example concerns estimation of a log likelihood function that can be
written as
,where , , are independent observed data, and
*X* is an unobserved variable.
Estimation of via direct simulation or importance sampling
is very inefficient because is much more concentrated
than *f*_{X}; simple use of the proposed method makes
the simulation very efficient.

*Keywords:* control variate, importance sampling, importance
sampling weight diagnostics, likelihood estimation, ratio estimate,
regression estimate, surface estimation, variance reduction

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