Past exam of the simulation and statistical programming course of the University of Oxford / 2023 / 1 / c / i

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- Solution i

Solution

Draw independent and identically distributed random variables

Y_{1}, \dots, Y_{n} \sim q

and use the importance weight

w = p / q

. The importance sampling estimator is

θ_{n}^{IS} = \frac{1}{n} i = 1 \sum n w (Y_{i}) ϕ (Y_{i}) .

The envelope supplies the support condition for importance sampling. Since

E_{q} [w (Y) ϕ (Y)] = E_{p} ϕ (X) = θ

, this is an unbiased estimator. Moreover,

0 \leq w \leq M

, so

E_{q} [w (Y)^{2} ϕ (Y)^{2}] = \int w (x) ϕ (x)^{2} p (x) d x = E_{p} [w (X) ϕ (X)^{2}] \leq M E_{p} [ϕ (X)^{2}] = M (1 + θ^{2}) .

Here

E_{p} ϕ^{2} = Var_{p} (ϕ) + θ^{2} = 1 + θ^{2}

. Subtracting the squared expected value and dividing by

n

proves the bounded-weight importance-sampling moment bound in the requested form:

Var (θ_{n}^{IS}) = \frac{E _{p} [ w ϕ ^{2} ] - θ ^{2}}{n} \leq \frac{M + ( M - 1 ) θ ^{2}}{n} .

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