Chapter 4 BART

4.1 A BART version of our hierachical trees model

We have a variable of interest for which we assume:

$$\begin{equation} y_{ij} = \sum_{k = 1}^{m} \overbrace{\mathbb{G}}^\text{Tree look up function}(\underbrace{X_{ij}}_\text{Covariates}, \overbrace{T_{k}}^\text{Tree structure}, \overbrace{M_{k}}^\text{Terminal node parameters}) + \underbrace{\epsilon_{ij}}_\text{Noise} \end{equation}$$

for observation $i = i, \dots, n_j$ in group $j = 1, \dots, a$. We also have that:

$$\begin{equation} \epsilon_{ij} \sim N(0, \tau^{-1}), \end{equation}$$

where $\tau^{-1}$ is the residual precision. In this setting, $M_{k}$ will represent the terminal node parameters + the individual group parameters for tree $k$.

For a single terminal node, let:

$$\begin{equation} R_{ijk1} = Y_{ij}^{(1)} - \sum_{t \neq k} \mathbb{G}(X_{ij}^{(1)}, T_{t}, M_{t}) \end{equation}$$

which represents the partial residuals for observation i, in group j, for tree k in terminal node 1. Now, let

$$\begin{equation} \underset{\sim}{R_j} = \{R_{ij}, \dots, j = 1,\dots, a \} \end{equation}$$

then

$$\begin{equation} \underset{\sim}{R_j} \sim N(\mu_j, \tau^{-1}), \\ \mu_j \sim N(\mu, k_1\tau^{-1}/m), \text{(m = number of trees)} \\ \mu \sim N(0, k_2 \tau^{-1}/m)\\ \end{equation}$$

using the same marginalisation as for a single tree:

$$\begin{equation} \underset{\sim}{R_j} \sim MVN(\mu \mathbf{1}, \tau^{-1} (k_1 M^{-1}MM^{T} + \mathbb{I})), \text{(M = group model matrix)}\\ \text{using the same trick as before and } \Psi = k_1 M^{-1}MM^{T} + \mathbb{I}: \\ \underset{\sim}{R_j} \sim MVN(0, \tau^{-1} (\Psi + k_2 M^{-1} \mathbf{1}\mathbf{1}^{T})), \end{equation}$$

which is used yo get the marginal distribution of a new tree. The new posterior updates will be:

$$\begin{equation} \mu | \dots \sim MVN( \frac{\mathbf{1}^{T} \Psi^{-1} R }{\tau \Psi^{-1} \mathbf{1} + k_2^{-1} M^{-1}}, \tau^{-1} (\mathbf{1}^{T} \Psi^{-1} \mathbf{1} + k_2^{-1} M^{-1})), \end{equation}$$

$$\begin{equation} \mu_j | \dots \sim MVN( \tau^{-1} (n_j + k_1^{-1} M^{-1})) \end{equation}$$

The update for $\tau$ will be a little different. Let $\hat f_{ij}$ be the overall prediction for observation $R_{ij}$ at the current iteration:

$$\begin{equation} \tau | \dots \sim Ga( \frac{n + m + 1}{2} + \alpha, \frac{\sum_{i, j} (y_{ij} - \hat f_{ij})^2}{2} + \frac{\sum_{j, k} (\mu_{jk} - \mu_{k})^2}{2} + \frac{\sum_{j, k} \mu_{k}^2}{2} + \beta ) \end{equation}$$