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This is my sampling algorithm

Let p(x) be a discrete distribution and f(x) be some function on real numbers. Consider

  1. $\textbf{Generate Samples from } p(x): \text{Initially, you sample } x_1, x_2, \ldots, x_n \text{ from the probability distribution } p(x).$ $ \text{Sample } x_i \text{ from } p(x), \text{ for } i = 1, 2, \ldots, n. $

  2. $\textbf{Reweigh Samples Using } f(x): \text{After generating these samples, reweigh them using the weights } \frac{f(x_i)}{\sum_{j=1}^n f(x_j)}$ $ \text{Weight for each sample } x_i: \frac{f(x_i)}{\sum_{j=1}^n f(x_j)}. $ Resample a point from this sampled set based on these weights.

What would be the marginal distribution of a point obtained using these two steps? I feel that it should be $\frac{p(x)f(x)}{\sum_{x'}p(x')f(x')}$ but I can't prove it?

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The distribution of a point given all the first stage samples is $$ g(X = x_i \mid x_1, \ldots, x_n) = \frac{f(x_i)}{\sum_{j=1}^n f(x_j)}. $$ So the marginal distribution for finite $n$ is $$ \int \cdots \int g(X = x_i \mid x_1, \ldots, x_n)p(x_1) \cdots p(x_n) dx_1 \cdots dx_n. $$

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It's not quite that, because your actual weight distribution (as written) has only the observed values in the denominator, not all values.

To take the extreme case, suppose $n=1$. You sample one value, $x_1$, from $p(x)$ and you reweight it using the weight $f(x_1)/f(x_1)=1$, so the marginal distribution is just $p(x)$

As $n$ increases, the distribution gets closer to being proportional to $p(x)f(x)$ because $$\frac{\sum_{i=1}^n f(x_j)}{\sum_{x'}f(x')}\stackrel{p}{\to} 1$$ by the law of large numbers

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