Notation Reference

Variables and Values

Symbol	Meaning
$X,Y,Z$	Random variables (capital letters)
$x,y,z$	Specific realized values of the corresponding random variables
$\mathbf{x},\mathbf{y}$	Vectors (bold lowercase); e.g. $\mathbf{x}\in {\mathbb{R}}^d$
$\mathbf{X}$	A matrix (bold uppercase); e.g. design matrix $\mathbf{X}\in {\mathbb{R}}^{N\times d}$

Probability

Symbol	Meaning
$p(x)$	Probability mass function (PMF) or Probability density function (PDF) evaluated at $x$ — for discrete/continuous variables
$p(X)$	Probability distribution of random variable $X$
$p(y\mid x)$	Conditional PMF/PDF of $Y=y$ given $X=x$
$p(Y\mid X)$	Conditional distribution of $Y$ given $X$
$p(x,y)$	Join PDF/PMF of $p(X,Y)$ at $X=x$ and $Y=y$
$p(X,Y)$	Join distribution of $X$ and $Y$
$p(\mathbf{x})$	join PDF/PMF over vector $\mathbf{x}$
$p_{\theta }(\mathbf{x})$	join PDF/PMF over vector $\mathbf{x}$, parametrized by $\theta$
$p_{\theta }(\mathbf{x}\mid \mathbf{y})$	join PDF/PMF over vector $\mathbf{x}$ given $\mathbf{y}$, parametrized by $\theta$
$p(\mathcal{X})$	Join distribution of all possible $\mathbf{x}\in \{{\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(N)}\}$

Datasets and Expectations

Symbol	Meaning
$\mathcal{X}=\{{\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(N)}\}$	Dataset of $N$ i.i.d. samples
${\mathbb{E}}_{\mathbf{x}\sim p(\mathbf{x})}[f(\mathbf{x})]$	Expectation of $f(\mathbf{x})$ under distribution $p$
$\mathbf{x}\stackrel{\text{i.i.d.}}{\sim }p(\mathbf{x})$	Samples drawn independently and identically from $p(\mathcal{X})$

Common Distributions

Symbol	Meaning
$\mathcal{N}(\mu ,{\sigma }^2)$	Univariate Gaussian with mean $\mu$ and variance ${\sigma }^2$
$\mathcal{N}(\mathbf{\mu },\mathbf{\Sigma })$	Multivariate Gaussian with mean $\mathbf{\mu }$ and covariance $\mathbf{\Sigma }$
$\mathcal{N}(0,\mathbf{I})$	Standard multivariate Gaussian (zero mean, identity covariance)
$\text{Bernoulli}(p)$	Bernoulli distribution with success probability $p$