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If I'm understanding rejection sampling correctly, it's a way for us to sample a distribution that is difficult to directly sample. In order to apply rejection sampling of a distribution $p(z)$, we set $p(z) = {1 \over Z_p}\tilde{p}(z)$. From there we create a proposal distribution $q(z)$ and find a constant $k$ such that $kq(z) \ge \tilde{p}(z)$.

What I'm confused about is $\tilde{p}(z)$. In the book (PR and ML by Bishop), it says that we can easily able to evaluate $p(z)$ for any given value of z up to some normalizing constant $Z$. What are some cases of this? I can't think of a reason why we need to use $\tilde{p}(z)$ when we know $p(z)$.

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  • $\begingroup$ Sampling from the univariate truncated normal distribution: link.springer.com/article/10.1007%2FBF00143942?LI=true $\endgroup$ – user3903581 Jul 16 '17 at 8:54
  • $\begingroup$ There are many cases (as in Bayesian statistics) when the density to simulate is known up to a normalising constant. $\endgroup$ – Xi'an Mar 18 '18 at 17:21
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Using $\tilde{p}(z)$ sometimes makes the math easier to work out because you can ignore the normalizing constant $Z$. One example is drawing from a beta(2,2) via rejection sampling. You can bound the density by 2, or you can drop the normalizing constant $\frac{\Gamma(4)}{\Gamma(2)\Gamma(2)}$ and work directly with the kernel, $p(1-p)$.

Or, in the notation of the OP, $\tilde{p}(z) = z(1-z)$ and $p(z) =z(1-z) \frac{\Gamma(4)}{\Gamma(2)\Gamma(2)}$

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