Chapter 2 Model

Define the following model. Suppose we have the observation of a tree node as: $y_{ij}, i = 1,\ldots,n_j, \; j = 1\ldots, m$ where $$y_{ij}$$ is observation $$i$$ in group $$j$$. There are different numbers of observations $$n_j$$ in each group, so for example $$n_1$$ is the number of observations in group 1, etc. There are $$m$$ groups. The total number of observations is $$n = \sum_{j=1}^m n_j$$

Then, for each tree node, suppose we have the likelihood: $y_{ij} \sim N(\mu_j, \tau^{-1})$

so each group has an overall mean $$\mu_j$$, with an overall precision term $$\tau$$.

We then have a hierarchical prior distribution:

$\mu_j \sim N(\mu, k_1 (\tau^{-1}))$

where $$k_1$$ will be taken as a constant for simplicity, and the hyper-parameter prior distributions are:

$\mu \sim N(0, \tau_{\mu} = k_2 (\tau^{-1}))$ $\tau \sim Ga(\alpha, \beta)$

Where the values $$k_1, k_2, \alpha, \beta$$ are all fixed.

2.1 Maths

• The likelihood of each tree node will be:

$$$L = \prod_{j = 1}^{m} \prod_{i = 1}^{n_j} p(y_{ij} | \mu_{j}, \tau) \\ L \propto \tau^{n/2} exp \{ -\frac{\tau}{2} \sum_{j = 1}^{m} \sum_{i = 1}^{n_j} (y_{ij} - \mu_j)^2 \}$$$

with prior distributions:

• $$\mu_j | \mu, \tau, k_1$$

$$$p(\mu_1, \dots, \mu_m | \mu, \tau) \propto (\tau/k_1)^{m/2} exp\{ - \frac{\tau}{2k_1} \sum_{j = 1}^{m} (\mu_j - \mu)^2 \}$$$

• $$\tau | \alpha, \beta$$

$p(\tau | \alpha, \beta) \propto \tau^{\alpha - 1} exp\{ - \beta \tau \}$

• $$\mu | \tau_{\mu} = k_2 (\tau^{-1})$$

$$$p(\mu | k_2, \tau) \propto (\tau/k_2)^{1/2} exp\{ - \frac{ \tau}{2 k_2} \mu^2 \} \}$$$

and their joint distribution as:

• $$p(\tau, \mu_1, \dots, \mu_m, \mu| y, k_1, k_2, \tau, \alpha, \beta)$$

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