$$ \begin{array}{lll} r_i & \sim & N(r_i | \mu_i, \sigma)\\ \mu_i & = & \mathbf{w} \cdot \mathbf{Q}_i\\ \mathbf{w} & \sim & N(\mathbf{w} | \boldsymbol{\phi}, \boldsymbol{\Sigma})\\ \pi(\boldsymbol{\phi}, \boldsymbol{\Sigma}) & = & NIW(0, 1, k+1, I_k) \end{array} $$ Where $\sigma$ and $\mathbf{Q}$ are known, but the latter is different at each step. The goal is to approximate $\pi_n(\boldsymbol{\phi}, \boldsymbol{\Sigma})$.