Chapter 4 Posteriors

4.1 Posterior for \(\tau\)

• \(p(\tau | \mu_1, \dots, \mu_m, y, \alpha, \beta, k_1)\)

\[\begin{equation} p(\tau | \mu_1, \dots, \mu_m, y, \alpha, \beta, k_1) \propto \tau^{\alpha - 1} exp\{ - \beta \tau \} \times \\ (\tau/k_1)^{m/2} exp \{ -\frac{\tau}{2k_1} \sum_{j = 1}^{m} (\mu_{j} - \mu)^2 \} \times \\ \tau^{n/2} exp \{ -\frac{\tau}{2} \sum_{j = 1}^{m} \sum_{i = 1}^{n_j} (y_{ij} - \mu_j)^2 \} \times (\tau/k_2)^{1/2} exp\{ - \frac{ \tau}{2 k_2} \mu^2 \} \\ \propto \tau^{(n+m+1)/2 + \alpha - 1 } exp \{ - \tau \Big( \frac{\sum_{j = 1}^{m} \sum_{i = 1}^{n_j} (y_{ij} - \mu_j)^2}{2} + \beta + \frac{\sum_{j = 1}^{m}(\mu_{j} - \mu)^2}{2k_1} + \frac{\mu^2}{2k_2}\Big) \} \end{equation}\]

So \(\tau | \mu_1, \dots, \mu_m, y, \alpha, \beta, k_1, k_2 \sim \\ \text{Gamma}((n+m+1)/2 + \alpha, \Big( \frac{\sum_{j = 1}^{m} \sum_{i = 1}^{n_j} (y_{ij} - \mu_j)^2}{2} + \beta + \frac{\sum_{j = 1}^{m}(\mu_{j} - \mu)^2}{2k_1} + \frac{\mu^2}{2k_2})\Big)\)

4.2 Posterior for \(\mu_j\)

\[\begin{equation} Q = (\tau/k_1) \sum_{j=1}^{m} (\mu_j - \mu)^2 + \tau \sum_{j = 1}^{m} \sum_{i = 1}^{n_j} (y_{ij} - \mu_j)^2 \\ Q = \tau [ \sum_{j = 1}^{m} n_j \mu_j^2 + \frac{\mu_j^2}{k_1} - (\sum_{j = 1}^{m} \frac{2 \mu \mu_j}{k_1} + 2 \bar y_j n_j \mu_j)] \\ Q \propto \tau [ \sum_{j = 1}^{m} (n_j + \frac{1}{k_1}) \mu_j^2 - 2 \mu_j (\frac{\mu}{k_1} + \bar y_j n_j )] \\ Q \propto [ \sum_{j = 1}^{m} (n_j + \frac{1}{k_1})(\mu_j - \frac{\mu/k_1 + \bar y_j n_j}{n_j + 1/k_1})^2]\\ \end{equation}\] So for each \(\mu_j\)

\[\mu_j | \mu, y, \tau, k_1 \sim N(\frac{\mu/k_1 + \bar y_j n_j}{n_j + 1/k_1}, ((n_j + \frac{1}{k_1}) \tau )^{-1})\]

4.3 Posterior for \(\mu\)

Similarly, for \(\mu\) we have:

\[\begin{equation} Q = \frac{\tau}{k_1} \sum_{j = 1}^{m} (\mu_j - \mu)^2 + \frac{ \tau}{k_2} \mu^2 \\ Q = \frac{\tau}{k_1} \sum_{j=1}^{m} (\mu_j^{2} - 2 \mu \mu_j + \mu^2) + \frac{ \tau}{k_2} \mu^2 \\ Q \propto (\frac{\tau}{k_2} + \frac{\tau}{k_1} m ) \mu^2 - \frac{2\tau}{k_1} \sum_{j=1}^{m} \mu \mu_j \\ Q \propto (\frac{\tau}{k_2} + \frac{\tau}{k_1} m ) \mu^2 - \frac{2\tau}{k_1} \mu \bar \mu m \\ Q \propto (\tau(\frac{m}{k_1} + \frac{1}{k_2}))(\mu - \frac{(\tau/k_1) \bar \mu m}{\tau(\frac{m}{k_1} + \frac{1}{k_2})})^2 \end{equation}\]

So for \(\mu\) we have that the conditional distribution:

\[\mu | \mu_1, \dots, \mu_{m}, \mu_{\mu}, k_1, k_2, \tau \sim N( \frac{(\tau/k_1) \bar \mu m}{\tau(\frac{m}{k_1} + \frac{1}{k_2})}, (\tau(\frac{m}{k_1} + \frac{1}{k_2}))^{-1})\]

4.4 A second posterior, with \(\mu_j\) marginalised out

The following is what is used in the code.

Assuming \[ y | \tau, k_1, k_2 \sim N(0, \tau^{-1}[(k_1 \mathbf{M}\mathbf{M}^{T} + \mathbf{I}) + k_2 \mathbf{1}\mathbf{1}^{T}])\]

we can do

\[\begin{equation} p(\mu | y, \alpha, \beta, k_1, k_2) \propto \exp \{ -\frac{\tau}{2}(\mathbf{y} - \mu \mathbf{1})^{T} \Psi^{-1} (\mathbf{y} - \mu \mathbf{1}) \} \times \\ \exp \{ -\frac{\tau}{2 k_2} \mu^{2} \} \\ \propto \exp \{ -\frac{\tau}{2}(\mu^{2} (\mathbf{1}^{T} \Psi^{-1} \mathbf{1} + 1/k_2) - 2 \mu \mathbf{1}^{T} \Psi^{-1} \mathbf{y} + \mu^2/k_2) \} \\ \propto \exp \{ -\frac{\tau}{2}[( \mathbf{1}^{T} \Psi^{-1} \mathbf{1}+ 1/k_2)( \mu^2 - \frac{2 \mu \mathbf{1}^{T} \Psi^{-1} \mathbf{y}}{\mathbf{1}^{T} \Psi^{-1} \mathbf{1} + 1/k_1}) \} \\ \end{equation}\]

\[\mu | y, \tau, k_1, k_2 \sim MVN(\frac{\mathbf{1}^{T}\Psi^{-1}\mathbf{y}}{\mathbf{1}^{T}\Psi^{-1}\mathbf{1} + k_2^{-1}}, ((\mathbf{1}^{T}\Psi^{-1}\mathbf{1} + k_2^{-1}) \tau )^{-1})\]