SIR explanation in Robert and Casella Intro to Monte Carlo Methods - How to do this derivation?

Question

Why is it an exact simulation from $f$, and not only an approximation?

I get $\begin{split} P(X^*\in A) & = \sum_i^n P(X^*\in A , X^* = X_i)=\sum_i^n P(X^*\in A | X^* = X_i)P(X^* = X_i) \\ & = \sum_i^n P(X_i\in A)\frac{f(X_i)}{ng(X_i)} \\ & = \sum_i^n 1_{A}(X_i)\frac{f(X_i)}{ng(X_i)} \rightarrow^P E(1_{A}(X))=P(X \in A) \end{split}=$

@whuber thanks for the comment, but could you be more explicit on the error I did? — An old man in the sea., Jul 30 '18 at 14:14
https://stats.stackexchange.com/questions/282889/confusion-conditioning-a-discrete-rv-on-a-continuous-rv-sampling-importance-r?rq=1 — An old man in the sea., Jul 30 '18 at 14:49
@Anoldmaninthesea. my answer that you linked to works if you just sum from $i=1$ to $n$. — Taylor, Jul 30 '18 at 17:35
@Taylor Yes, thanks for your answer. But I don't know what to think of the 3rd equality in your answer. You have a continuous variable for which $P(X=a)\neq 0$. You state it's not a probability measure, but then what is it? That's the only point that makes it hard to accept your answer. ;) — An old man in the sea., Jul 30 '18 at 17:44

Taylor · Accepted Answer · 2018-07-30T19:00:28.750

Maybe this notation is better? I write $g(a)$ for the density of $X_i$. Also $P(X^* = X_i \mid X_i = a)$ is a kernel. It's measurable and integrable in $a$.

\begin{aligned} P(X^* \in A) &= \sum_i P(X^* \in A \text{ and } X^* = X_i) \\ &= \sum_i P(X_i \in A \text{ and } X^* = X_i) \\ &= \sum_i\int_{ A} g(a_i)P(X^* = X_i \mid X_i = a_i)da_i \quad{*} \\ &= \sum_i\int_{ A} g(a_i)\frac{f(a_i)}{ng(a_i)}da_i\\ &= \int_{ A} f(a)da. \end{aligned}

Also, this is exact because $\sum_i \frac{f(x_i)}{ng(x_i)} = 1$. A lot of times you have to use the self-normalized weights $\frac{f(x_j)}{g(x_j)}\left[\sum_i \frac{f(x_i)}{g(x_i)}\right]^{-1}$.

Taylor, if you find the time, https://stats.stackexchange.com/questions/359977/conditional-expectation-on-an-estimator-for-defensive-sampling ;) — An old man in the sea., Jul 31 '18 at 11:28

SIR explanation in Robert and Casella Intro to Monte Carlo Methods - How to do this derivation?

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