LaTeX

R = 
\begin{pmatrix}
1 & r_0r & r_0r^2 & \cdots & r_0r^{T-1} \\
r_0r & 1 & r_0r & \cdots & r_0r^{T-2} \\
r_0r^2 & r_0r & 1 & \cdots & r_0r^{T-3} \\ 
\vdots & \vdots & \vdots & \vdots & \vdots \\ 
r_0r^{T-1} & r_0r^{T-2} & r_0r^{T-3} & \cdots & 1 
\end{pmatrix}

\[ R = \begin{pmatrix} 1 & r_0r & r_0r^2 & \cdots & r_0r^{T-1} \\ r_0r & 1 & r_0r & \cdots & r_0r^{T-2} \\ r_0r^2 & r_0r & 1 & \cdots & r_0r^{T-3} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ r_0r^{T-1} & r_0r^{T-2} & r_0r^{T-3} & \cdots & 1 \end{pmatrix} \]

s(\\boldsymbol{\\hat{\\theta}}, y) = 
\begin{pmatrix} 
s(\hat{\theta}, y_1)_1 & s(\hat{\theta}, y_1)_2 & \cdots & s(\hat{\theta}, y_1)_k \\
\vdots & \vdots & \ddots & \vdots \\
s(\hat{\theta}, y_n)_1 & s(\hat{\theta}, y_n)_2 & \cdots & s(\hat{\theta}, y_n)_k \\
\end{pmatrix}

\[ s(\boldsymbol{\hat{\theta}}, y) = \begin{pmatrix} s(\hat{\theta}, y_1)_1 & s(\hat{\theta}, y_1)_2 & \cdots & s(\hat{\theta}, y_1)_k \\ \vdots & \vdots & \ddots & \vdots \\ s(\hat{\theta}, y_n)_1 & s(\hat{\theta}, y_n)_2 & \cdots & s(\hat{\theta}, y_n)_k \\ \end{pmatrix} \]

[1 \; x_{1t} \; x_{2t} \; \cdots \; x_{mt}] 
\cdot \left( \begin{array}{ccc} 
\hat{\beta}_{01} & \cdots & \hat{\beta}_{0k} \\ 
\vdots & \ddots & \vdots \\ 
\hat{\beta}_{m1} & \cdots & \hat{\beta}_{mk} \end{array} \right) 
= 
[\hat{y}_{t1} \; \hat{y}_{t2} \; \cdots \; \hat{y}_{tk}]

\[ [1 \; x_{1t} \; x_{2t} \; \cdots \; x_{mt}] \cdot \left( \begin{array}{ccc} \hat{\beta}_{01} & \cdots & \hat{\beta}_{0k} \\ \vdots & \ddots & \vdots \\ \hat{\beta}_{m1} & \cdots & \hat{\beta}_{mk} \end{array} \right) = [\hat{y}_{t1} \; \hat{y}_{t2} \; \cdots \; \hat{y}_{tk}] \]

ICC_{tt} = ICC_{tt'} = \frac {\sigma_b^2} {\sigma_b^2 + \sigma_e^2} r_{tt'}

\[ ICC_{tt} = ICC_{tt'} = \frac {\sigma_b^2} {\sigma_b^2 + \sigma_e^2} r_{tt'} \]

y_{ict} = \mu + \beta_0t + \beta_1X_{ct} + b_{ct} + e_{ict}

\[ y_{ict} = \mu + \beta_0t + \beta_1X_{ct} + b_{ct} + e_{ict} \]

\hat{\theta} = \arg\max_{\theta \in \Theta} \sum\_{i=1}^n w_i^{\text{forest}} \\cdot l(\theta; y_i)

\[ \hat{\theta} = \arg\max_{\theta \in \Theta} \sum_{i=1}^n w_i^{\text{forest}} \cdot l(\theta; y_i) \]

\begin{align*}
H(Y|X_1,\cdots, X_k) = \sum_{d = 0,1} \sum_{{i_1}=1}^c \cdots \sum_{{i_k}=1}^c \\
P(y_d | x_{i_1}, \cdots, x_{i_k}) \log P(y_d | x_{i_1}, \cdots, x_{i_k})
\end{align*}

\[ \begin{align*} H(Y|X_1,\cdots, X_k) = \sum_{d = 0,1} \sum_{{i_1}=1}^c \cdots \sum_{{i_k}=1}^c \\ P(y_d | x_{i_1}, \cdots, x_{i_k}) \log P(y_d | x_{i_1}, \cdots, x_{i_k}) \end{align*} \]

\begin{align*}
x^2 + y^2 &= 1 \\
y &= \sqrt{1 - x^2}
\end{align*}

• With the “&” symbols, the 2nd line stays lined up with the end of the first line

\[ \begin{align*} x^2 + y^2 &= 1 \\ y &= \sqrt{1 - x^2} \end{align*} \]

\begin{tabular} {lll}
Transformation & Function & Elasticity \\
\hline
Level Level & Y\;=\;a+bX & \epsilon = b \cdot \frac {X} {Y} \\  
Log Level & \log Y = a+bx & \epsilon = b \cdot X \\
Level-Log & Y = a + b \cdot \log X & \epsilon = \frac {b} {Y} \\
Log-Log & \log Y = a + b \cdot \log X & \epsilon = b \\
\hline
\end{tabular}

• This is “Raw LaTeX” and will be ignored when the output is HTML. (See Docs)
• In the last column, for some unknown reason, using \frac without another “\” symbol before it throws an error. Adding \epsilon = to the expression made it work fine.

c_{max}(t,\mu, \sigma) = \max \limits_{k = 1,..., dim(T)} \left | \frac {(t-\mu)_k} {\sqrt {\Sigma_{kk}}} \right |

• Large absolute pipes require \left, \right
• The text underneath “max” requires \limits

\[ c_{max}(t,\mu, \sigma) = \max \limits_{k = 1,..., dim(T)} \left | \frac {(t-\mu)_k} {\sqrt {\Sigma_{kk}}} \right | \]

\begin{align*}
\mbox{net present value (npv)} = &\sum \limits_{i = 0}^m benefit_i*(1-discount)^i -\\
&\sum \limits_{i=0}^m cost_i*(1-discount)^i
\end{align*}

\[ \begin{align*} \mbox{net present value (npv)} = &\sum \limits_{i = 0}^m benefit_i*(1-discount)^i -\\ &\sum \limits_{i=0}^m cost_i*(1-discount)^i \end{align*} \]

\epsilon_Z \stackrel{\text{iid}}{\sim} \mathcal{N}(0, 0.1)

\[ \epsilon_Z \stackrel{\text{iid}}{\sim} \mathcal{N}(0, 0.1) \]

\mbox{posterior} = \frac {\mbox{Prob of observed variables} \;\times\; \mbox{Prior}} {\mbox{Normalizing constant}}

\[ \mbox{posterior} = \frac {\mbox{Prob of observed variables} \;\times\; \mbox{Prior}} {\mbox{Normalizing constant}} \]

\lim\limits_{x \to +\infty} \sup \frac{\bar{F}(x)}{\bar{F}_{\exp}(x)} = \frac{\bar{F}(x)}{e^{-\lambda x}}, \;\; \forall \lambda > 0

\[ \lim\limits_{x \to +\infty} \sup \frac{\bar{F}(x)}{\bar{F}_{\exp}(x)} = \frac{\bar{F}(x)}{e^{-\lambda x}}, \;\; \forall \lambda > 0 \]

\text{vec} = \left( \begin{array}{cc} \text{rand var1} \\
 \text {rand var2} \end{array} \right)

\[ \text{vec} = \left( \begin{array}{cc} \text{rand var1} \\ \text {rand var2} \end{array} \right) \]

SBD(x,y) = 1 - \frac {\max (\text{NCCc}(x,y))} {\left\lVert x \right\rVert_2 \; \left\lVert y \right\rVert_2}

\[ SBD(x,y) = 1 - \frac {\max (\text{NCCc}(x,y))} {\left\lVert x \right\rVert_2 \; \left\lVert y \right\rVert_2} \]

\begin{align}
&D \not\!\perp\!\!\!\perp A \\
&D \!\perp\!\!\!\perp A \\
&Y \!\perp\!\!\!\perp X|Z
&Y \perp\mkern-10mu\perp X\;|\;Z
\end{align}

\[ \begin{align} &D \not\!\perp\!\!\!\perp A \\ &D \!\perp\!\!\!\perp A \\ &Y \!\perp\!\!\!\perp X|Z \\ &Y \perp\mkern-10mu\perp X\;|\;Z \end{align} \]

||y-f||^2 + \lambda \int \left(\frac {\partial^2 f(\text{log[baseline profit]})}{\partial \; \text{log[baseline profit]}^2}\right)^2 \partial x

\[ ||y-f||^2 + \lambda \int \left(\frac {\partial^2 f(\text{log[baseline profit]})}{\partial \; \text{log[baseline profit]}^2}\right)^2 \partial x \]

E_{iz_{nm}} = 
\left\{ \begin{array}{lcl}
-\beta_z z_{zm}P_{nm} \left(1+ \frac{1-\lambda_k}{\lambda_k} \frac {1}{P_{nB_k}} \right) & \mbox{if} \; m \in \beta_k \\
-\beta_z z_{zm}P_{nm} & \mbox{if} \; m \in \beta_k 
\end{array}\right.

• There is no dash between {array} and {lcl}. It’s just two curly brackets touching each other
  • If you want the expressions on the left side to right-align use {rcl} — for center-align, leave it blank I think
• Do not forget that period at end — to the right of \right

\[ E_{iz_{nm}} = \left\{ \begin{array}{lcl} -\beta_z z_{zm}P_{nm} \left(1+ \frac{1-\lambda_k}{\lambda_k} \frac {1}{P_{nB_k}} \right) & \mbox{if} \; m \in \beta_k \\ -\beta_z z_{zm}P_{nm} & \mbox{if} \; m \notin \beta_k \end{array}\right. \]

\begin{align}
&P_{ni} = \int L_{ni}(\beta) \; f(\beta\;|\;\boldsymbol{\theta}) d\beta\\
&\mbox{where} \; L_{ni}(\beta) = \frac {e^{V_{ni}(\beta)}}{\sum_{j=1}^J e^{V_{ni}(\beta)}} 
\end{align}

• Alignment occurs where the & symbols are positioned
• \boldsymbol is used bold the vector \(\theta\)

\[ \begin{align} &P_{ni} = \int L_{ni}(\beta) \; f(\beta\;|\;\boldsymbol{\theta}) \; d\beta\\ &\mbox{where} \; L_{ni}(\beta) = \frac {e^{V_{ni}(\beta)}}{\sum_{j=1}^J e^{V_{ni}(\beta)}} \end{align} \]

\pi_i = \mbox{Pr}(i \in S) = \sum \limits_{i \in s \in S} \mbox{Pr}(S = s) = \frac {\binom{N-1}{n-1}}{\binom{N}{n}} = \frac {n}{N}

• \binom used for binomial coefficient/combination notation

\[ \pi_i = \mbox{Pr}(i \in S) = \sum \limits_{i \in s \in S} \mbox{Pr}(S = s) = \frac {\binom{N-1}{n-1}}{\binom{N}{n}} = \frac {n}{N} \]

\begin{aligned}
&R_{i,j} = \frac{S_i + S_j}{M_{i,j}}\\
&\begin{aligned}
\text{where} \;\; &S_i = \left(\frac{1}{T_i} \sum_{j=1}^{T_i} \lVert X_j - A_i \rVert_{p}^q \right)^{\frac{1}{q}} \;\; \text{and} \\
&M_{i,j} = \lVert A_i - A_j \rVert_p = \left(\sum_{k=1}^n |a_{k,i} - a_{k,j}|^p\right)^{\frac{1}{p}}
\end{aligned}
\end{aligned}

• The key for this nested align is the “&” before the second \begin{aligned}. Otherwise “where” and \(R_{i,j}\) won’t be aligned flushly on the left side.

\[ \begin{aligned} &R_{i,j} = \frac{S_i + S_j}{M_{i,j}}\\ &\begin{aligned} \text{where} \;\; &S_i = \left(\frac{1}{T_i} \sum_{j=1}^{T_i} \lVert X_j - A_i \rVert_{p}^q \right)^{\frac{1}{q}} \;\; \text{and} \\ &M_{i,j} = \lVert A_i - A_j \rVert_p = \left(\sum_{k=1}^n |a_{k,i} - a_{k,j}|^p\right)^{\frac{1}{p}} \end{aligned} \end{aligned} \]

\begin{aligned}
&Y_{ij} = [\alpha_0 + \beta_0 \text{large}_{ij}] + [u_i + v_i \text{large}_{ij} + \epsilon_{ij}] \\
&\begin{aligned}
\text{where}\quad \epsilon &\sim \mathcal{N}(0, \sigma^2) \: \text{and}\\
\left[ \begin{array}{cc} u_i \\ v_i \end{array} \right] &\sim \mathcal{N} \left(\left[\begin{array}{cc} 0\\0 \end{array}\right], \left[\begin{array}{cc} \sigma^2_u\\\rho\sigma_u \sigma_v \quad \sigma^2_v \end{array}\right]\right)
\end{aligned}
\end{aligned}

\[ \begin{aligned} &Y_{ij} = [\alpha_0 + \beta_0 \text{large}_{ij}] + [u_i + v_i \text{large}_{ij} + \epsilon_{ij}] \\ &\begin{aligned} \text{where}\quad \epsilon &\sim \mathcal{N}(0, \sigma^2) \: \text{and}\\ \left[ \begin{array}{cc} u_i \\ v_i \end{array} \right] &\sim \mathcal{N} \left(\left[\begin{array}{cc} 0\\0 \end{array}\right], \left[\begin{array}{cc} \sigma^2_u\\\rho\sigma_u \sigma_v \quad \sigma^2_v \end{array}\right]\right) \end{aligned} \end{aligned} \]

\left[\begin{array}{cc} u_i\\v_i\\w_i\\x_i\\y_i\\z_i \end{array} \right] 
\sim \mathcal{N} \left(
\left[\begin{array}{cc} 0\\0\\0\\0\\0\\0\end{array} \right],
\begin{bmatrix}
\sigma_u^2 \\
\sigma_{uv} & \sigma_v^2 \\
\sigma_{uw} & \sigma_{vw} & \sigma^2_w \\
\sigma_{ux} & \sigma_{vx} & \sigma_{wx} & \sigma_x^2 \\
\sigma_{uy} & \sigma_{vy} & \sigma_{wy} & \sigma_{xy} & \sigma_{y}^2 \\
\sigma_{uz} & \sigma_{vz} & \sigma_{wz} & \sigma_{xz} & \sigma_{yz} & \sigma_z^2
\end{bmatrix}
\right)  

• bmatrix stands for bracket matrix. Previous matrices (above) used pmatrix which stands for parentheses matrix.
• Others:
  • Bmatrix for curly braces matrix
  • vmatrix for a matrix with “rectangular line boundary”
  • Vmatrix for a matrix with double vertical bars
  • matrix for a matrix without brackets

\[ \left[\begin{array}{cc} u_i\\v_i\\w_i\\x_i\\y_i\\z_i \end{array} \right] \sim \mathcal{N} \left( \left[\begin{array}{cc} 0\\0\\0\\0\\0\\0\end{array} \right], \begin{bmatrix} \sigma_u^2 \\ \sigma_{uv} & \sigma_v^2 \\ \sigma_{uw} & \sigma_{vw} & \sigma^2_w \\ \sigma_{ux} & \sigma_{vx} & \sigma_{wx} & \sigma_x^2 \\ \sigma_{uy} & \sigma_{vy} & \sigma_{wy} & \sigma_{xy} & \sigma_{y}^2 \\ \sigma_{uz} & \sigma_{vz} & \sigma_{wz} & \sigma_{xz} & \sigma_{yz} & \sigma_z^2 \end{bmatrix} \right) \]

\begin{align}
&\ \textbf{Registered provinces for INGO } i \\
\text{Count of provinces}\ \sim&\ \operatorname{Ordered\,Beta}(\mu_{i_j}, \phi_y, k_{0_y}, k_{1_y}) \\[8pt]
&\textbf{Model of Outcome Average} \\
\mu_i = &\
  \mathrlap{\begin{aligned}[t]
  &\beta_0 + \beta_1\ \text{Issue[Arts and Culture]} + \beta_2\ \text{Issue[Education]}\ + \\
  &\beta_3\ \text{Issue[Industry and Humanitarian]} + \beta_4\ \text{Issue[Economy and Trade]}
  \end{aligned}}\\[8pt]
&\ \textbf{Priors} \\
\beta_0\ \sim&\ \operatorname{Student\,t}(\nu = 3, \mu = 0, \sigma = 2.5) && \text{Intercept} \\
\beta_{1..12}\ \sim&\ \mathcal{N}(0, 5) && \text{Coefficients}
\end{align}

• textbf is bold font text
• \operatorname treats text as functions like \max or \sin in terms of spacing
• \mathrlap + \begin{aligned}allows you to wrap extra long equations

\[ \begin{align} &\ \textbf{Registered Provinces for INGO } i \\ \text{Count of Provinces}\ \sim&\ \operatorname{Ordered\,Beta}(\mu_{i_j}, \phi_y, k_{0_y}, k_{1_y}) \\[8pt] &\textbf{Model of Outcome Average} \\ \mu_i = &\ \mathrlap{\begin{aligned}[t] &\beta_0 + \beta_1\ \text{Issue[Arts and Culture]} + \beta_2\ \text{Issue[Education]}\ + \\ &\beta_3\ \text{Issue[Industry and Humanitarian]} + \beta_4\ \text{Issue[Economy and Trade]} \end{aligned}}\\[4pt] &\ \textbf{Priors} \\ \beta_0\ \sim&\ \operatorname{Student\,t}(\nu = 3, \mu = 0, \sigma = 2.5) && \text{Intercept} \\ \beta_{1..12}\ \sim&\ \mathcal{N}(0, 5) && \text{Coefficients} \end{align} \]

\begin{align}
  \text{Transition}_{\text{2-step}} &= 
  \begin{aligned}[t]
    &P(X_{n-1} = 0|X_{n-2} = 0) \cdot P(X_n = 1|X_{n-1} = 0) \\
    &+ P(X_{n-1} = 1|X_{n-2} = 0) \cdot P(X_n = 1|X_{n-1} = 1) \\
    &+ P(X_{n-1} = 2|X_{n-2} = 0) \cdot P(X_n = 1|X_{n-1} = 2)
  \end{aligned}\\
  &= (0.7 \cdot 0.2) + (0.3 \cdot 0.4) + (0 \cdot 0) = 0.33
\end{align}

• [t] says this nested aligned section should be top aligned with “Transition” and not centered on “Transition.”

\[ \begin{align} \text{Transition}_{\text{2-step}} &= \begin{aligned}[t] &P(X_{n-1} = 0|X_{n-2} = 0) \cdot P(X_n = 1|X_{n-1} = 0) \\ &+ P(X_{n-1} = 1|X_{n-2} = 0) \cdot P(X_n = 1|X_{n-1} = 1) \\ &+ P(X_{n-1} = 2|X_{n-2} = 0) \cdot P(X_n = 1|X_{n-1} = 2) \end{aligned}\\ &= (0.7 \cdot 0.2) + (0.3 \cdot 0.4) + (0 \cdot 0) = 0.33 \end{align} \]