Chapter 4 New Marginal Distributions (y)

Suppose we have the outcome variable: \[\mathbf{y} \sim MVN_n(\mu \mathbf{1}_n, \tau^{-1} (k_1 \mathbf{MM}^T + \mathbf{I}))\]

with: \[\mu \sim N(\mu_\mu, \tau_\mu^{-1})\]

And define \(\mathbf{\Psi} = k_1 \mathbf{MM}^T + \mathbf{I}\)

Then, as a ‘trick’ to estimate the mean and variance of \(\mathbf{y}\), we can write:

\[\mathbf{y} = \mu \mathbf{1}_n + \left[ \tau^{-1} \Psi \right]^{1/2} \mathbf{z}\]

where \(\mathbf{z} \sim MVN(0, \mathbf{I})\) is a standard multivariate normal. Then we have:

\[E(\mathbf{y}) = \mu_\mu \mathbf{1}_n + 0 = \mu_\mu \mathbf{1}_n \\ Var(\mathbf{y}) = Var(\mu \mathbf{1}_n) + Var(\left[\tau^{-1} \Psi \right]^{1/2}) = \tau_{\mu}^{-1} \mathbf{1}_n \mathbf{1}^T_n + \tau^{-1} \mathbf{\Psi}\]

Now let \(\tau_\mu^{-1} = k_2 \tau^{-1}\), we get:

\[\mathbf{y} \sim MVN(\mu_\mu \mathbf{1}, \tau^{-1} \left[ k_2 \mathbf{1}_n \mathbf{1}^T_n + \mathbf{\Psi} \right]) \equiv MVN(W_0, \tau^{-1} W_1)\]

We now want to marginalize this over \(\tau \sim Ga(\alpha, \beta)\), by integrating out a Gamma distribution with:

\[ Ga\Big(n/2 + \alpha, \beta + \frac{(\mathbf{y} - \mathbf{W}_0)^T \mathbf{W}_1^{-1} (\mathbf{y} - \mathbf{W}_0)}{2}\Big) \]

…so we get:

\[\pi(\mathbf{y} | \mu_\mu, k_1, k_2) = \int \tau^{n/2} | \mathbf{W}_1 |^{-1/2} \exp \left[ -\frac{\tau}{2} (\mathbf{y} - \mathbf{W}_0)^T \mathbf{W}_1^{-1} (\mathbf{y} - \mathbf{W}_0)\right] \tau^{\alpha - 1} \exp(-\beta \tau) \partial \tau\]

This becomes:

\[ = | \mathbf{W}_1 |^{-1/2} \Gamma \left( \frac{n}{2} + \alpha \right) \left[ \frac{(\mathbf{y} - \mathbf{W}_0)^T \mathbf{W}_1^{-1} (\mathbf{y} - \mathbf{W}_0)}{2} + \beta\right]^{-\left(\frac{n}{2} + \alpha \right)}\]

(For examples of the evaluation of this marginal distribution, see )

4.0.1 log version of the marginal:

This equation in log-scale gives us:

\[\begin{eqnarray*} \log(\pi(\boldsymbol{y} | k_1, k_2, \mu, \alpha, \beta)) &=& -\frac{1}{2} \log(|\boldsymbol{\mathbf{W}}_{1}|) + \log(\Gamma(N/2 + \alpha)) \\ &-& (N/2 + \alpha)\left[ \log \beta + \log\Big(\frac{(\mathbf{y} - \mathbf{W}_{0})^T \mathbf{W}_{1}^{-1} (\mathbf{y} - \mathbf{W}_{0})}{2}\Big) \right] \end{eqnarray*}\]

And the same, when written for \(j = 1,\dots, b\) nodes of a tree, would look like:

\[\begin{eqnarray*} \sum_{j = 1}^{b} \log(\pi(\boldsymbol{y_{j}} | N_j, k_1, k_2, \mu, \alpha, \beta)) &=& \sum_{j = 1}^{b} -\frac{1}{2} \log(|\boldsymbol{\mathbf{W}}_{1,j}|) + \log(\Gamma(N_j/2 + \alpha)) \\ &-& (N_j/2 + \alpha)\left[ \log \beta + \log\Big(\frac{(\mathbf{y}_j - \mathbf{W}_{0,j})^T \mathbf{W}_{1,j}^{-1} (\mathbf{y}_j - \mathbf{W}_{0,j})}{2} \Big) \right] \end{eqnarray*}\]