Vectors are denoted by lower case bold Roman letters such as $\textbf{x}$, and all vectos are assumed to be column vectors.
A superscript $T$ denotes the transpose of a matrix or vector, so that $\textbf{x}^T$ will be a row vector.
Uppercase bold roman letters, such as $\textbf{M}$, denote matrices.
The notation $(w_1,…,w_M)$ denotes a row vector with $M$ elements, while the corresponding column vector is written as $\textbf{w} = (w_1, …, w_M)^T$.
The notation $[a,b]$ is used to denote the closed interval from a to b, that is the interval including the values a and b themselves, while $(a, b)$ denotes the corresponding open interval, that is the interval excluding a and b.
The $M \times M$ identity matrix (aslo know as the unit matrix) is denoted $\text{I}_M$, which will be abbreviated to $\textbf{I}$ where there is no ambiguity about it dimensionality.
The notation of $g(x) = O(f(x))$ denotes that $|f(x)/g(x)|$ is bounded as $x \rightarrow \infty$. For instance if $g(x) = 3x^2+2$, then $g(x)=O(x^2)$.
The expectation of a function $f(x,y)$ with respect to a random variable $x$ is denoted by $E_{x}[f(x,y)]$. In situations where there is no ambiguity as to which varaible is being averaged over, this will be simplified by omitting the suffix, for instance $E[x]$. If the distribution of $x$ is conditioned on another variable $z$, then the corresponding conditional expectation will be written $E_{x}[f(x|z)]$. Similarly, the variance is denoted $\text{var}[f(x)]$, and for vector variables the convariance is written $\text{cov}[\textbf{x}, \textbf{y}]$. We shall also use $\text{cov}[\textbf{x}]$ as a shorthand notation for $\text{cov}[\textbf{x}, \textbf{x}]$.
If we have $N$ values $x_1,…,x_n$ of D-dimensional vectors $\textbf{x} = (x_1, …,x_D)^T$, we can conbine the observations into a data matrix $\textbf{X}$ in which the $n^{th}$ row of $\textbf{X}$ corresponds to the row vector $x_n^T$. Thus the $n, i$ element of $\textbf{X}$ corresponds to the $i^{th}$ element of the $n^{th}$ observation $\textbf{x}_n$. For the case of one-dimensional varaibles we shall denote such a matrix by $\mathbf{x}$, which is a column vector whose $n^{th}$ element is $x_n$. Note that $\mathbf{x}$(which has dimensionality $N$) usea a different typeface to distinguish it from $\textbf{x}$ (which has dimensionality $D$).