by izhak
Last Updated March 14, 2019 20:19 PM

What is the purpose of envelope function?

Say, for single random variable, why can't we just sample x coordinate and y coordinate (representing probability) and reject all those y's which lay above the curve?

How do you propose sampling $y$? You take a sample from some other distribution $q$. What happens if $q(x) < p(x)$ where p is the desired distribution? You cannot have an acceptance ratio greater than one and so the samples would not be distributed according to $p$.

Thus you need a function $q$ where $q$ is greater than $p$ for all $x$ but is close to $p$ so as to minimise rejection. This might then resemble an envelope.

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