master.tex

\begin{comment}
  #+ORGTBL: SEND Conjugates orgtbl-to-latex :splice nil :skip 0 :no-escape t
  |----------------------+---------------------------------+-------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------+--------------------------------------------------------------------|
  | L                    | P                               | ConjP                                                                         | Posterior                                                                                     | Predictive                                                         |
  |----------------------+---------------------------------+-------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------+--------------------------------------------------------------------|
  | \textsc{Binomial}    | $\theta$                        | $\textsc{Beta}(a, b)$                                                         | $a + y$, $b + n - y$                                                                          | $\textsc{BetaBinomial}(y)$                                         |
  | \textsc{Poisson}     | $\theta$                        | $\textsc{Gamma}(a, b)$                                                        | $a + y$, $b + n$                                                                              | $\textsc{NegativeBinomial}(y)$                                     |
  | \textsc{Normal}      | $\mu$                           | $\textsc{Normal}(\gamma, \frac{\sigma^{2}}{n_{0}})$                           | $\frac{n_{0} \gamma + n \overline y}{n_0 + n}, \sigma^{2}_{n} = \frac{\sigma^{2}}{n_{0} + n}$ | $\textsc{Normal}(\gamma_{n}, \sigma^{2} + \sigma^{2}_{n})$         |
  | \textsc{Normal}      | $\mu$                           | $\textsc{Normal}(\gamma, \tau_{0})$ (precision)                               | $\frac{\tau_{0} \gamma + n \overline y}{\tau_{0} + n \tau}, \tau_{n} = \tau_{0} + n \tau$     | $\textsc{Normal}(\gamma_{n}, \frac{1}{\tau_{n}} + \frac{1}{\tau})$ |
  | \textsc{Normal}      | $\sigma^{2} = \frac{1}{\omega}$ | $\omega \sim \textsc{Gamma}(\frac{n_{0}}{2}, \frac{n_{0} \sigma_{0}^{2}}{2})$ | $\frac{n_{0} + n}{2}, \frac{n_{0} \sigma_{0}^{2}}{2} + \frac{1}{2} \sum (y_{i} - \mu)^{2}$    |                                                                    |
  | \textsc{Multinomial} | $p_{1}, \dots, p_{k}$           | $\textsc{Dirichlet}(\alpha_{1}, \dots, \alpha_{k})$                           | $\alpha_{1} + n_{1}, \dots, \alpha_{k} + n_{k}$                                               |                                                                    |
  |----------------------+---------------------------------+-------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------+--------------------------------------------------------------------|
\end{comment}

\begin{comment}
  #+ORGTBL: SEND Distributions orgtbl-to-latex :splice nil :skip 0 :no-escape t
  |-----------------------------------------------------+------------------------------------------------------------------------------+------------------------------------------+-----------------------------------|
  | Distribution                                        | Density                                                                      | Mean                                     | Variance                          |
  |-----------------------------------------------------+------------------------------------------------------------------------------+------------------------------------------+-----------------------------------|
  | $\textsc{Normal}(\mu, \sigma^{2})$                  | $\frac{1}{\sqrt{2 \pi \sigma^{2}}} exp(-\frac{(x - \mu)^{2}}{2 \sigma^{2}})$ | $\mu$                                    | $\sigma^{2}$                      |
  | $\textsc{Poisson}(\lambda)$                         | $\frac{e^{-\lambda} \lambda^{k}}{k!}$                                        | $\lambda$                                | $\lambda$                         |
  | $\textsc{Gamma}(a, b)$                              | $\frac{b^{a}}{\Gamma(a)} x^{a-1} e^{-bx}$                                    | $\frac{a}{b}$                            | $\frac{a}{b^{2}}$                 |
  | $\textsc{Beta}(a, b)$                               | $\frac{\Gamma(a + b)}{\Gamma(a) \Gamma(b)} x^{a-1}(1-x)^{b-1}$               | $\frac{a}{a+b}$                          | $\frac{ab}{(a+b)^{2}(a + b + 1)}$ |
  | $\textsc{Dirichlet}(\alpha_{1}, \dots, \alpha_{K})$ | $\propto \prod_{i=1}^{K} x_{i}^{\alpha_{i} - 1}$                             | $\frac{\alpha_{i}}{\sum_{k} \alpha_{k}}$ |                                   |
  |-----------------------------------------------------+------------------------------------------------------------------------------+------------------------------------------+-----------------------------------|
\end{comment}