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Suppose we want a sample of size n from a truncated gaussian distribution with density

$f(x) = \dfrac{1}{\sqrt{2\pi}\sigma(1-\Phi((1-\mu )/\sigma ) }e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ , $x>1$

set

$p(x) = \dfrac{f(x)}{(1-\Phi((1-\mu )/\sigma )^{-1}}$

Then we can find an envelope for $p(x)$ which is equal to the density of the normal distribution with mean $\mu$ and variance $\sigma^2$.

For the rejection sampling method - just sample from a normal distribution with mean $\mu$ and variance $\sigma^2$ and throw away any sample $y$ that is less than or equal to one.

What is the difference between SIR and Rejection sampling in this case?

(SIR) - generate samples from $f$.

1.draw samples $x^1..x^N \sim h(x)$

2,calculate importance weights $w_i = f(x^i)/h(x^i)$

3,Normalize the weights as $q_i = w_i/\sum_j w_j$

4, resample from $\{x^1,..x^N \}$ where $y^i$ is drawn with probability $q_i$.

we can exchange $f(x^i)$ by $p(x^i)$ in 2 , since it will cancel in 3.

As in the rejection sampling we use the envelope $h(x) = p(x)$ and so we resample from the set in $4$, all with equal probabilities. But there are no approximation here since all draws are good that is all x in (4) come from a normal distribution.. and all that are less Than or equal to one we must throw away

Is my reasoning Wrong?

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  • $\begingroup$ What do you mean by "all draws are good"? The justification for SIR is asymptotic, so your draws arein fact approximately distributed according to the target. $\endgroup$ – Taylor Jan 3 '18 at 0:03
  • $\begingroup$ I mean that any draw that we pick in 4 still come from a normal distribution and we keep only those that are greater than one and so the ones we keep come from an trunc normal distribution ...@Taylor $\endgroup$ – Danny Jan 3 '18 at 0:15
  • $\begingroup$ So why is that not exact $\endgroup$ – Danny Jan 3 '18 at 0:33
  • $\begingroup$ @Taylor: the justification for Monte Carlo is equally asymptotic, aka LLN. $\endgroup$ – Xi'an Jan 3 '18 at 4:28
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In this special case, there is no difference: simulating a sample $x_1,\ldots,x_N$ from the $\mathcal{N}(\mu,\sigma^2)$ distribution will produce the same truncated sample $y_1,\ldots,y_n$ from the truncated Gaussian distribution because the importance weights are then zero or one.

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