Question

Suppose that we wish to approximate the integral\(\int g (x)dx\). Suppose that we have a p.d.f. \(f\)that we shall use as an importance function. Suppose that \(g(x)/f(x)\) is bounded. Prove that the importance sampling estimator has finite variance.

Short answer

Directly follows from the fact that the quotient is bounded.

Step-by-step solution

Definition for importance sampling

• Many integrals can be advantageously recast as random variable functions.
• We can estimate integrals that would otherwise be impossible to compute in closed form if we can simulate a large number of random variables with proper distributions.

Determine the importance sampling estimator

Assume that the quotient is bounded.

Since \(f\) is the importance function, it follows that

\({Y^{(i)}} = \frac{{g\left( {{X^{(i)}}} \right)}}{{f\left( {{X^{(i)}}} \right)}}\)Are random variables from which the importance sampling estimator is obtained.

The importance sampling estimator is

\(Z = \frac{1}{\nu }\sum\limits_{i = 1}^\nu {{Y^{(i)}}} \)

And the p.d.f. of \({X^{(i)}}\)is \({f_i}\) thus, because the quotient

\(Y(X) = \frac{{g(X)}}{{f(X)}}\)

Is bounded, the variance of \(Z\) is finite.