Appendix A — Notation

Below are mathematical notational rules used throughout the course.

• Scalar quantities as denoted as italicized symbols, such as \(x\), \(y\), \(\mu\), and \(\sigma\).
• Vector quantities (first-rank tensors) are denoted in bold, such as \(\mathbf{x}\), \(\mathbf{y}\), \(\boldsymbol{\mu}\), and \(\boldsymbol{\sigma}\).
• Matrix quantities (second-rank tensors) are denoted with sans serif capital letters, such as \(\mathsf{A}\), \(\mathsf{W}\), and \(\mathsf{\sigma}\).
• The one exception to the boldface and sans serif convention is when we denote a generic set of data or parameters. In that case, we use standard italicized symbols like \(\theta\) (typically for a set of parameters) or \(z\) (typically for a set of latent variables).
• Subscripts typically denote an element of a vector, such as \(x_i\), or an element of a matrix, such as \(A_{ij}\). They can also denote an entry in a non-ordered collection, such as \(M_i\).
• Transposes are denoted with a superscript \(\mathsf{T}\).
• Vector dot products result in a scalar and are denoted with a dot, such as \(\mathbf{x}\cdot\mathbf{y}\). Note that this is denoted as \(\mathbf{x}^\mathsf{T}\mathbf{x}\) in some texts, but we will not use that notation. Writing out the sum, this is

\[\begin{aligned} \mathbf{x}\cdot\mathbf{y} = \sum_{i}x_i\, y_i. \end{aligned} \tag{A.1}\]

• Matrix-vector products result in a vector are also denoted with a dot, such as \(\mathsf{A}\cdot\mathbf{x}\). Writing out the sum, this is

\[\begin{aligned} \mathsf{A}\cdot\mathbf{x} = \begin{pmatrix}\sum_{i}A_{i1} x_i \\ \sum_{i}A_{i2} x_i \\ \vdots \end{pmatrix} \end{aligned} \tag{A.2}\]

• Matrix-matrix multiplication results in a matrix and is also denoted with a dot, such as \(\mathsf{A}\cdot\mathsf{B}\).
• We denote probability mass functions or probability density functions of measured quantities with \(f\), such as \(f(y\mid \theta)\). We denote probability mass functions or probability density functions of parameters or unmeasured quantities with \(g\), such as \(g(\theta \mid y)\). We denotes PMFs or PDFs of mixed or unknown variables with \(\pi\), such as \(\pi(y, \theta)\) or \(\pi(z)\).