# 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 $$q$$