> [!note] Notation > Lowercase letters $f,p,q,$ etc. are pdfs or pmfs. > > Uppercase letters $F,P,Q$, etc. are cdfs. $\Phi$ is the standard normal cdf. > > Bold letters $\mathbf{X}$ are vector/multi-dimensional random variables. > > Variables with hats, e.g. $\hat{\theta}$, are estimates for those without, e.g. $\theta$. > > $X \sim p$ is the variable we want to simulate. > > $Y \sim q$ are proposal distributions when relevant. The proposal is sometimes written as $X \overset{?}{=}y$. > > $U$ are uniform variables, most likely $U[0,1]$. > > Subscripts in expectations like $\mathbb{E}_{X}$ specifies the expectation to be integrated over the density of $X \sim p$, as opposed to say $\mathbb{E}_{Y}$ with $Y \sim q$: $\begin{align*} \mathbb{E}_{X}[\phi(X)]&= \int p(x)\phi(x) \, dx,\\ \mathbb{E}_{Y}[\phi(Y)]&= \int q(y)\phi(y) \, dy, \end{align*}$which are in general different quantities. ### Topics ![[NotesWithDomain.base#NonSnippets]] ### Snippets ![[NotesWithDomain.base#Snippets]]