sampling

Common sampling notation

$X\sim P$ — “$X$ is distributed as $P$”.
$x_i\stackrel{\text{iid}}{\sim }P,i=1,\dots ,n$ — iid samples.
$x^{(i)}\sim P$ — Monte Carlo sample $i$ (common in ML/VI).
Algorithmic pseudocode: $x\leftarrow \mathrm{Sample}(P)$ or $x\leftarrow P$.
Conditional/controlled: $a_t\sim {\pi }_{\theta }(\cdot \mid s_t)$ (RL).
Evo/ES style: $x_k^{(t)}\sim \mathcal{N}(m^{(t)},C^{(t)})$, population $\{x_k^{(t)}{\}}_{k=1}^{\lambda }$.

“Reparametrization trick”

Given a mean (vector) $\mu$ and a standard deviation (vector) $\sigma$, and a (vector of) noise $z$ from a spherical standard gaussian $z\sim \mathcal{N}(0,I)$, we can sample from the gaussian with mean $\mu$ and standard deviation $\sigma$ by scaling and shifting the noise:

$$x=\mu +\sigma \odot z$$

where $\odot$ denotes the element-wise product.

Sampling $k$ elements from $n$ choices

When order matters:

With replacement: $\#=n^k$
Without replacement: $\#=\frac{n!}{(n-k)!}=n(n-1)\dots (n-k+1)$ (pool gets smaller with each sample)

When order doesn’t matter:
$\#=\frac{n!}{(n-k)!k!}=\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ (binomial coefficient = “selection from a partial multiset”)

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