Latex Reference

2. Latex Reference#

I’m going to experiment with this auto-generated Latex reference based on all the macros used in this Jupyter Book. You can see the code used to generate it at this Github repo. To use any of the macros below, just put a backslash (\) in front of them. One other useful thing I want to add is the align environment for aligning equations. Here’s an example:

\begin{align*}
	y_i | \theta, X &\sim \text{Poisson}(\lambda_i) \\
	\lambda_i &= g^{-1}(\eta_i) \\
	\eta_i &= \beta_0 + \sum_{j=1}^p \beta_j x_{ij} \\
	\beta &\sim \mathcal{N}\left(0, \sigma^2\right)
\end{align*}					

This displays:

\[\begin{split} \begin{align*} y_i | \theta, X &\sim \text{Poisson}(\lambda_i) \\ \lambda_i &= g^{-1}(\eta_i) \\ \eta_i &= \beta_0 + \sum_{j=1}^p \beta_j x_{ij} \\ \beta &\sim \mathcal{N}\left(0, \sigma^2\right) \end{align*} \end{split}\]

Once you’ve opened align, you just put an & symbol before the character you want to align on. Then end each line with two slashes: \\. The * after align suppresses line numbering.

You can also add a second aligned item per-line. Say you wanted to add comments to your equation to explain each step:

\begin{align*}
f(y | \theta) &= \prod_{i=1}^{n} e^{l_i} && \text{Log-likelihood to likelihood.}\\
&= \prod_{i=1}^{n} \frac{e^{-(-l_i)}(-l_i)^0}{0!} && \text{Matching the form of the Poisson PMF.} \\
&= \prod_{i=1}^{n} f_P(0; -l_i) && \text{Poisson evaluated at zero with mean $-l_i$.}
\end{align*}

This displays:

\[\begin{split} \begin{align*} f(y | \theta) &= \prod_{i=1}^{n} e^{l_i} && \text{Log-likelihood to likelihood.}\\ &= \prod_{i=1}^{n} \frac{e^{-(-l_i)}(-l_i)^0}{0!} && \text{Matching the form of the Poisson PMF.} \\ &= \prod_{i=1}^{n} f_P(0; -l_i) && \text{Poisson evaluated at zero with mean $-l_i$.} \end{align*} \end{split}\]

The remainder of this document was automatically created by OpenAI’s GPT-4 on July 15, 2023.

Greek Letters#

alpha, $\alpha$, represents the lowercase Greek letter alpha. Example: $\alpha = 0.05$.
beta, $\beta$, represents the lowercase Greek letter beta. Example: $\beta = 0.5$.
gamma, $\gamma$, represents the lowercase Greek letter gamma. Example: $\gamma = 0.7$.
delta, $\delta$, represents the lowercase Greek letter delta. Example: $\delta = 0.1$.
theta, $\theta$, represents the lowercase Greek letter theta. Example: $\theta = 1.2$.
lambda, $\lambda$, represents the lowercase Greek letter lambda. Example: $\lambda = 2.3$.
xi, $\xi$, represents the lowercase Greek letter xi. Example: $\xi = 3.4$.
pi, $\pi$, represents the lowercase Greek letter pi. Example: $\pi \approx 3.14159$.
sigma, $\sigma$, represents the lowercase Greek letter sigma. Example: $\sigma = 0.25$.
tau, $\tau$, represents the lowercase Greek letter tau. Example: $\tau = 0.5$.
phi, $\phi$, represents the lowercase Greek letter phi. Example: $\phi = 1.61803$.
Delta, $\Delta$, represents the uppercase Greek letter Delta. Example: $\Delta = 5$.
Gamma, $\Gamma$, represents the uppercase Greek letter Gamma. Example: $\Gamma(1) = 1$.
Sigma, $\Sigma$, represents the uppercase Greek letter Sigma. Example: $\Sigma x_i = \sum_{i=1}^{n} x_i$.
Phi, $\Phi$, represents the uppercase Greek letter Phi. Example: $\Phi(x)$ is the cumulative distribution function of a standard normal distribution.
Nu, $\nu$, represents the lowercase Greek letter nu. Example: $\nu$ is often used to denote degrees of freedom in statistics.

Operators#

arg, $\arg$, represents the argument of a complex number. Example: $\arg(z)$.
binom, $\binom{n}{k}$, represents the binomial coefficient. Example: $\binom{5}{2} = 10$.
sqrt, $\sqrt{x}$, represents the square root of x. Example: $\sqrt{4} = 2$.
frac, $\frac{a}{b}$, represents the fraction a divided by b. Example: $\frac{1}{2} = 0.5$.
max, $\max$, represents the maximum value. Example: $\max\{1, 2, 3\} = 3$.
int, $\int$, represents the integral symbol. Example: $\int x dx = \frac{1}{2}x^2$.
prod, $\prod$, represents the product symbol. Example: $\prod_{i=1}^{n} x_i$.
sum, $\sum$, represents the summation symbol. Example: $\sum_{i=1}^{n} x_i$.

Symbols#

infty, $\infty$, represents the symbol for infinity. Example: $\lim_{x\to\infty} f(x)$.
ldots, $\ldots$, represents the symbol for ellipsis. Example: $1, 2, 3, \ldots, n$.
approx, $\approx$, represents the symbol for approximately equal to. Example: $\pi \approx 3.14$.
sim, $\sim$, represents the symbol for distributed as. Example: $X \sim N(\mu, \sigma^2)$.
propto, $\propto$, represents the symbol for proportional to. Example: $y \propto x$.
mid, $\mid$, represents the symbol for such that. Example: $\{x \mid x > 0\}$.
neq, $\neq$, represents the symbol for not equal to. Example: $1 \neq 2$.
in, $\in$, represents the symbol for element of. Example: $x \in \mathbb{R}$.
mapsto, $\mapsto$, represents the symbol for mapping. Example: $f: x \mapsto x^2$.

Formatting#

text, \text{your text here}, allows you to insert normal text into math mode. Example: $x = 2$ \text{ and } $y = 3$.
mathbf, $\mathbf{x}$, makes the letter bold. Example: $\mathbf{x}$ is often used to denote a vector.
hat, $\hat{x}$, puts a hat over the letter. Example: $\hat{x}$ is often used to denote an estimate.
bar, $\bar{x}$, puts a bar over the letter. Example: $\bar{x}$ is often used to denote the sample mean.
mathcal, $\mathcal{A}$, creates a calligraphic letter. Example: $\mathcal{A}$ is often used to denote a set.
quad, \quad, creates a space in math mode. Example: $a = b \quad \text{and} \quad c = d$.
left and right, $\left(\frac{a}{b}\right)$, creates scalable delimiters. Example: $\left(\frac{a}{b}\right)$.

Comparisons#

lt, $<$, represents the less than symbol. Example: $1 < 2$.
le, $\leq$, represents the less than or equal to symbol. Example: $1 \leq 2$.
gt, $>$, represents the greater than symbol. Example: $3 > 2$.
ge, $\geq$, represents the greater than or equal to symbol. Example: $3 \geq 2$.

Arrows#

rightarrow, $\rightarrow$, represents the rightward arrow. Example: $x \rightarrow y$.
mapsto, $\mapsto$, represents the mapsto arrow. Example: $f: x \mapsto x^2$.
nabl, $\nabla$, represents the del or nabla symbol, used in vector calculus. Example: $\nabla f$.

Functions#

log, $\log$, represents the logarithm function. Example: $\log_{2}(8) = 3$.
arctan, $\arctan$, represents the inverse tangent function. Example: $\arctan(1) = \frac{\pi}{4}$.

Others#

times, $\times$, represents the multiplication symbol. Example: $2 \times 3 = 6$.
cdot, $\cdot$, represents the dot product symbol. Example: $\mathbf{a} \cdot \mathbf{b}$.
inf, $\inf$, represents the infimum or greatest lower bound of a set. Example: $\inf\{x \mid x > 0\} = 0$.