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In previous file, we derive the posterior assuming \(\Sigma\) is full-rank. We try to relax this constrain in this file.

\[\begin{align} x &\sim N_R(b, \Sigma) \\ b &\sim N_R(0, \Omega) \end{align}\]

Let \(VDV^\intercal\) be eigendecomposition of \(\Sigma\), \(V = [V_1 \quad V_2]\), \(V_1 \in \mathbb{R}^{R\times k}\), \(V_2 \in \mathbb{R}^{R\times R-k}\), where \(k\) is the rank of residual variance. \(D = diag(D_1, D_2 = 0)\), \(D_1\) is of size \(k\), \(D_2\) is of size \(R-k\). Note that \(V_1 D_1 V_1^\intercal = \Sigma\).

We have \[\begin{align} \tilde{x}_1 = V_1^\intercal x &\sim N(V_1^\intercal b, D_1), \\ \tilde{b}_1 = V_1^\intercal b &\sim N(0, V_1^\intercal \Omega V_1), \\ \tilde{x}_2 = V_2^\intercal x &\sim N(V_2^\intercal b, 0), \\ \tilde{b}_2 = V_2^\intercal b &\sim N(0, V_2^\intercal \Omega V_2). \end{align}\]

Note that \(V_1^\intercal \Omega V_1\) may not be full-rank, so we use result from file.

The posterior for \(\tilde{b}_1 \in \mathbb{R}^{k}\) is \[\begin{align} \tilde{b}_1|\tilde{x}_1 &\sim N_{k}( \mu_1 , \Sigma_1), \\ \Sigma_1 &= V_1^\intercal \Omega V_1 (I + D_1^{-1} V_1^\intercal \Omega V_1 )^{-1} \\ &= V_1^\intercal \Omega (I + \Sigma^\dagger \Omega)^{-1} V_1 \quad (\dagger \text{ means pseudo inverse}), \\ \mu_1 &= \Sigma_1 D_1^{-1} \tilde{x}_1. \end{align}\] The posterior for \(\tilde{b}_2 \in \mathbb{R}^{R-k}\) is a point mass on \(\tilde{x}_2\). We can write it as \[\begin{align} \tilde{b}_2|\tilde{x}_2 \sim N(\tilde{x}_2, 0). \end{align}\]

Therefore the posterior for \(V^\intercal b\) is \[ V^\intercal b | V^\intercal x \sim N\left( \left( \begin{matrix} \Sigma_1 D_1^{-1} V_1^\intercal x \\ V_2^\intercal x \end{matrix}\right), \left( \begin{matrix} \Sigma_1 & 0 \\ 0 & 0 \end{matrix} \right) \right). \]

The posterior for \(b\) is \[\begin{align} b | V^\intercal x &\sim N\left( V \left( \begin{matrix} \Sigma_1 D_1^{-1} V_1^\intercal x \\ V_2^\intercal x \end{matrix}\right), V \left( \begin{matrix} \Sigma_1 & 0 \\ 0 & 0 \end{matrix} \right) V^\intercal \right) \\ &\sim N\left( V_1 V_1^\intercal \Omega(I + \Sigma^\dagger \Omega)^{-1} \Sigma^\dagger x + V_2 V_2^\intercal x, V_1 V_1^\intercal \Omega (I + \Sigma^\dagger \Omega)^{-1} V_1 V_1^\intercal \right) \end{align}\]