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+\[\begin{align} + +& \textbf{ Intercept-only model } \\ + +& P\left ( \lambda, \mu, \phi \right ) \prod_{m=1}^{M} P\left ( \sigma_{m} | \mu_{m}, \lambda, \phi \right ) + \prod_{n=1}^{N} \prod_{m=1}^{M} P\left ( k_{m,n} | n_{n}, \mu_{m}, \sigma_{m} \right ) & \textrm{ (1) } \\ + +& k_{m,n} \sim \mathit{BetaBinomial^{*}} \left(n_{n}, \mu_{m}, \sigma_{m} \right ); \sum_{m=1}^{M} \mu_{m} = 0 & \textrm{ (2) } \\ + +& \sigma_{m} \sim \mathit{Normal} \left(\lambda_{0} + \lambda * \mu_{m}, \phi \right ) & \textrm{ (3) } \\ + +& \lambda \sim \mathit{Normal} \left(0,5 \right); \nu \sim \mathit{Normal} \left(0,2 \right); & \textrm{ (4) } \\ +\\ +& \textbf{ Regression model } \\ +\\ +& \mu_{m,n} = \left( \gamma_{m,c} * X_{c,n} \right ); \textrm{For c}\in[1..C] \sum_{m=1}^{M} \gamma_{m, c} = 0 & \textrm{ (5) } \\ +\\ + +& \textbf{ Beta-binomial reparametrisation } \\ +\\ +& \mathit{BetaBinomial^{*}} \left(n_{n}, \mu_{m}, \sigma_{m} \right ) & \textrm{ Alternative parametrisation (6) } \\ + +& \pi = \mathit{softmax} \left(\mu \right); = \frac{exp(\mu)}{\sum_{m=1}^{M} exp(\mu_{m}) } & \textrm{ (7) } \\ +& \omega = e^{\sigma} & \textrm{ (8) } \\ + +& \alpha_{m} = \pi_{m} * \omega_{m}; \beta_{m} = \left(1 - \pi_{m} \right) * \omega_{m}; & \textrm{ (9) } \\ + +& \mathit{BetaBinomial} \left(n_{n}, \alpha_{m}, \beta_{m} \right ) & \textrm{ Classic parametrisation (10) } \\ + + + + +\end{align}\]
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