# Symbols

# Symbols¶

\(\log(x)\) Natural logarithm of x

\(\mathbb{R}\) Real numbers

\(\mathbb{R}^n\) n-dimensional vector space of real numbers

\(\mathcal{A, S}\) Sets

\(x \in A\) Set membership. \(x\) is an element of the set \(A\)

\(\unicode{x1D7D9}_A\) Indicator function. Returns 1 if \(x \in A\) and 0 otherwise

\(a \propto b\) a is proportional to b

\(a \underset{\sim}{\propto} b\) a is approximately proportional to b

\(a \approx b\) a is approximately equal to b

\(a, c, \alpha, \gamma\) Scalars are lowercase

\(\mathbf{x, y}\) Vectors are bold lowercase, thus we write a column vector as \(\mathbf{x}=[x_1,\dots,x_n]^T\)

\(\mathbf{X, Y}\) Matrices are bold uppercase

\(X, Y\) Random variables are specified as upper case Roman letters

\(x, y\) Outcomes from random variables are generally specified as lower case roman letters

\(\boldsymbol{X, Y}\) Random vectors are in uppercase slanted bold font, \(\boldsymbol{X} = [X_1,\dots,X_n]^T\)

\(\boldsymbol{\theta}\) Greek lowercase characters are generally used for model parameters. Notice, that as we are Bayesians parameters are generally considered random variables

\(\hat \theta\) Point estimate of \(\boldsymbol{\theta}\)

\(\mathbb{E}_{X}[X]\) Expectation of \(X\) with respect to \(X\), most often than not this is abbreviated as \(\mathbb{E}[X]\)

\(\mathbb{V}_{X}[X]\) Variance of \(X\) with respect to \(X\), most often than not this is abbreviated as \(\mathbb{V}[X]\)

\(X \sim p\) Random variable \(X\) is distributed as p

\(p(\cdot)\) Probability density or probability mass function

\(p(y \mid \boldsymbol{x})\) Probability (density) of \(y\) given \(\boldsymbol{x}\). This is the short form for \(p(Y=y \mid \boldsymbol{X}=\boldsymbol{x})\)

\(f(x)\) An arbitrary function of x

\(f(\boldsymbol{X}; \theta, \gamma)\) \(f\) is a function of \(\boldsymbol{X}\) with parameters \(\theta\) and \(\gamma\). We use this notation to highlight that \(\boldsymbol{X}\) is the data we pass to a function or model and \(\theta\) and \(\gamma\) are parameters

\(\mathcal{N}(\mu, \sigma)\) A Gaussian (or normal) distribution with mean \(\mu\) and standard deviation \(\sigma\)

\(\mathcal{HN}(\sigma)\) A Half-Gaussian (or half-normal) distribution with standard deviation \(\sigma\)

Beta\((\alpha, \beta)\) Beta distribution with shape parameters \(\alpha\), \(\beta\)

Expo\((\lambda)\) An Exponential distribution with rate parameter \(\lambda\)

\(\mathcal{U}(a, b)\) A Uniform distribution with lower boundary \(a\) and upper boundary \(b\)

T\((\nu, \mu, \sigma)\) A Student’s t-distribution with grade of normality \(\nu\) (also known as degrees of freedom), location parameter \(\mu\) (the mean when \(\nu > 1\)), scale parameter \(\sigma\) (the standard deviation as \(\lim_{\nu\to\infty}\)).

\(\mathcal{H}\text{T}( \nu \sigma)\) A Half Student’s t-distribution with and grade of normality \(\nu\) (also known as degrees of freedom) and scale parameter \(\sigma\)

Cauchy\((\alpha, \beta)\) Cauchy distribution with location parameters \(\alpha\) and scale parameter \(\beta\)

\(\mathcal{H}\text{C}(\beta)\) Half-Cauchy distribution with scale parameter \(\beta\)

\(\text{Laplace}(\mu, \tau)\) Laplace distribution with mean \(\mu\) and scale \(\tau\)

Bin\((n, p)\) Binomial distribution with trials \(n\) and success \(p\)

Pois(\(\mu)\) Poisson distribution with mean (and variance) \(\mu\)

NB(\(\mu, \alpha)\) Negative Binomial distribution with Poisson parameter \(\mu\) and Gamma distribution parameter \(\alpha\)

\(\mathcal{G}RW(\mu, \sigma)\) Gaussian random walk distribution with innovation drift \(\mu\) and innovation standard deviation \(\sigma\)

\(\mathbb{KL}(p \parallel q)\) Kullback-Leibler divergence from \(p\) to \(q\)