Last Updated: February 11, 2024 - Added section for non-linear Bayesian models

Notation

Throughout, we’ll use capital letters to denote random variables, e.g. $X$. We will use boldface or vector-bar notation to denote vectors, e.g. $\mathbf x$ or $\vec{x}$. A random vector will therefore capital and bolded (or vector-barred), e.g. $\mathbf X$ or $\vec{X}$.

• Let $X_1, …, X_n$ be $n$ random variables, and $\mathbf X = (X_1, …., X_n)$ be a random vector.
• Sometimes we shall partition $\mathbf X = (\mathbf Y, \mathbf Z)$ into two or possibly more sub-vectors, $Y$ and $Z$.
• Given any vector, $\mathbf x$, if we do not indicate elsewhere, assume that it’s components are $\mathbf x = (x_1, …, x_m)$.
• We’ll tend to use greek letters for model parameters.

$\LaTeX$ Examples

$\\mathbf X \\sim \\mathcal N(\\boldsymbol \\mu, \\Sigma)$

$$ \mathbf X \sim \mathcal N(\boldsymbol \mu,\Sigma) $$

\\begin{align*}
y 
&= f(\\mathbf x) + W \\\\
&= \\theta_1 \\varphi_1(\\mathbf x) + \\cdots + 
   \\theta_n \\varphi_n(\\mathbf x) + W \\\\
&= \\sum_{i=1}^n \\theta_i \\varphi_i(\\mathbf x) + W
\\end{align*}

$$ \begin{align*} y &= f(\mathbf x) + W \\ &= \theta_1 \varphi_1(\mathbf x) + \cdots + \theta_n \varphi_n(\mathbf x)+W \\ &= \sum_{i=1}^n \theta_i \varphi_i(\mathbf x) + W\end{align*} $$

$$
A = 
\\begin{pmatrix}
1 & 2 & 3 \\\\
3 & 4 & 5 \\\\
8 & 9 & 10
\\end{pmatrix}
$$

$$ A = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 8 & 9 & 10 \end{pmatrix} $$

See also $\LaTeX$ cheat sheet: https://wch.github.io/latexsheet/. Additionally, you can click any displayed math on the course website to see its $\LaTeX$ encoding.

Random Variables and Distributions

CDF