 make the following approximation:

-   When assimilating a non-baseline observation, ${\bar{\mathcal{A}}}$ is kept fixed at its previous value, and for updating ${\mathcal{A}}$ it is assumed that ${\bar{\nu}}^{-1}=0$.

Approximation B2 implies that for ${\mathbf{x}}_t\neq{\mathbf{0}}$, $$\begin{aligned}
  Y_t|~ {\bar{\nu}},~ {\boldsymbol{\mu}},~ \nu,~ {\bar{\mathcal{A}}}_{t-1},~ {\mathcal{A}}_{t-1} & \stackrel{apx.}{\sim} 
      \mathrm{N} \left(~{\bar{m}}_{t-1} + {\mathbf{x}}_t^\top{\boldsymbol{\mu}},~ \nu^{-1}{\mathbf{x}}_t^\top{\mathbf{1}}~ \right), $$ which, given distributions at time $t-1$ as specified in , leads to the desired tractable updates for the sufficient statistics for as follows: ${\bar{\mathcal{A}}}_t = {\bar{\mathcal{A}}}_{t-1}$ and $$\begin{aligned}
  {\mathbf{m}}_t &= {\mathbf{m}}_{t-1} + q_t C_{t-1}{\mathbf{x}}_t(y_t - {\bar{m}}_{t} - {\mathbf{x}}_t^\top{\mathbf{m}}_{t-1}) &
  C_t  &= C_{t-1} - q_t C_{t-1} {\mathbf{x}}_t {\mathbf{x}}_t^\top C_{t-1} \nonumber\\
  a_t  &= a_{t-1} + \frac{1}{2} &
  b_t  &= b_{t-1} + \frac{q_t}{2}(y_t - {\bar{m}}_{t} - {\mathbf{x}}_t^\top{\mathbf{m}}_{t-1})^2, $$ where $q_t=({\mathbf{x}}_t^\top{\mathbf{1}}+ {\mathbf{x}}_t^\top C_{t-1} {\mathbf{x}}_t)^{-1}$. In essence, the approximate observation process decouples the learning about the observational parameters: when no MU fires then $({\bar{\mu}},{\bar{\nu}})$ is updated, else $(\mu,\nu)$ is updated.

After assimilating the baseline observations $y_1,\dots,y_{\tau-1}$, both ${\bar{\nu}}^{-1}$ and ${\bar{\mu}}$ are known (and known to be small) with considerable certainty. Thus, approximating ${\bar{\nu}}^{-1}$ as $0$ and considering ${\bar{\mu}}$ to be a point mass at ${\bar{m}}$ is reasonable. Furthermore, the prior for $\nu$ does not need to be set until just before the observation $y_{\tau}$ is assimilated. Given the tight posterior for ${\bar{\nu}}$ at this juncture it is, therefore, possible to incorporate the knowledge that ${\bar{\nu}}>>\nu$ into the vague prior for $\nu$ (which is conceptually equivale