support of an importance sampling with respect to the original distribution function

Question

I have a question regarding the support of an importance sampling distribution with respect to the support of the original distribution function. I was reading that the support of the importance sampling distribution (say the importance distribution is denoted q(x), say the set Q contains all value of x which satisfies q(x)>0 ) has to cover the support of the original distribution (say the distribution denoted as p(x) and let D denoted the set of x which satisfies p(x)>0 ), so it means we have $$D \subset Q$$.

The goal is to estimate some integral say:

$$\int_{D}f(x)p(x)dx$$

by using this instead:

$$\int_{Q} f(x) \frac{p(x)}{q(x)} q(x) dx $$

My question is that the author is saying q(x) doesn't have to be entirely bigger than zero as long as it is bigger than zero when $f(x)p(x)\neq 0$, i.e. $ x \in{Q}$ whenever $f(x)p(x) \neq 0$.

And next it goes about saying this:

$E_{q} \Bigg(\frac{f(X)p(x)}{q(x)} \Bigg)$ = $\int_{Q} \frac{f(x)p(x)}{q(x)} q(x) dx$ =
$\int_{D} f(x)p(x) dx$ + $\int_{Q \cap D^c} f(x)p(x) dx-\int_{D \cap Q^c} f(x)p(x) dx $

My question has to do with the very last line here, I don't really get why the expression on the very last line here would equal this:

$\int_{Q} \frac{f(x)p(x)}{q(x)} q(x) dx$.

It seems it is saying the set $Q$ is a union of the two sets: $D$ and $Q\cap D^c$. And $D\cap Q^c$ is the intersection of $D$ and $Q\cap D^c$.

My question is, is my understanding of the union and intersection correct regarding the above?

Because it seems it is based on the formula of:

$P(A\cup B) = P(A) + P(B) - P(A \cap B)$

If it is indeed based on the above simple "set union operation", then my question is how does the terms:

$\int_{D} f(x)p(x) dx$ + $\int_{Q \cap D^c} f(x)p(x) dx-\int_{D \cap Q^c} f(x)p(x) dx $

equal $\int_{Q} \frac{f(x)p(x)}{q(x)} q(x) dx$.

P.S. Hi Xian, yes it is here: http://statweb.stanford.edu/~owen/mc/Ch-var-is.pdf

It is just some notes I read from the pdf file.

Answer 1

This can be easily resolved via the use of (set) indicator functions:

My question is that the author is saying $q(x)$ doesn't have to be entirely bigger than zero as long as it is bigger than zero when $f(x)p(x)≠0$, i.e. $x∈Q$ whenever $f(x)p(x)≠0$.

Indeed, the only constraint for the identity to hold is that the support $Q$ of $q$ contains the support of the function $x \to f(x)p(x)$, because, if $H$ denotes this support$$\int_D f(x)p(x) dx=\int_D f(x)p(x)\mathbb{I}_{H}(x) dx=\int_{D\cap H} f(x)p(x) dx=\int_H f(x)p(x) dx$$

As for the second part, if $D\subset Q$, \begin{align} \int_{Q} f(x)p(x) dx&=\int_{Q} \{\mathbb{I}_D(x)+\mathbb{I}_{D^c}(x)\} f(x)p(x) dx\\ &=\int_{Q \cap D} f(x)p(x) dx+\int_{Q \cap D^c} f(x)p(x) dx \end{align} but otherwise, since $$\mathbb{I}_{Q\cup D}=\mathbb{I}_Q+\mathbb{I}_{D\cap Q^c}=\mathbb{I}_D+\mathbb{I}_{Q\cap D^c}$$ we have $$\mathbb{I}_Q=\mathbb{I}_D+\mathbb{I}_{Q\cap D^c}-\mathbb{I}_{D\cap Q^c}$$