Prior:
$$p_0(\theta_a) \varpropto \text{exp}(-\frac{1}{2}\theta_{a}^{T} (\lambda I_d)^{-1} \theta_a) = \text{exp}(-\frac{1}{2} \frac{1}{\lambda}\theta_{a}^{T} \theta_a)$$
MLE:
\begin{align*}
p(y_{t,a} | \theta_a) &= (2 \pi)^{\frac{n_{t,a}}{2}} |\sum|^{\frac{1}{2}} \text{exp}(-\frac{1}{2} (y_{t,a} - D_{t,a} \theta_{a})^{T} I_{n_{t,a}}^{-1} (y_{t,a} - D_{t,a} \theta_a)) \\
& \varpropto \text{exp}(-\frac{1}{2}(y_{t,a} - D_{t,a} \theta_{a})^{T} I_{n_{t,a}}^{-1} (y_{t,a} - D_{t,a} \theta_a))
\end{align*}
Hence,
\begin{align*}
p_{t,a}(\theta_a) &\varpropto \text{exp}(-\frac{1}{2} \frac{1}{\lambda} \theta_{a}^{T} \theta_a) ~ \text{exp}(-\frac{1}{2}(y_{t,a} - D_{t,a} \theta_{a})^{T} I_{n_{t,a}}^{-1} (y_{t,a} - D_{t,a} \theta_a)) \\
&\varpropto \text{exp}(-\frac{1}{2} (\frac{1}{\lambda} \theta_{a}^{T} \theta_a + y_{t,a}^{T} y_{t,a} - y_{t,a}^{T} D_{t,a} \theta_a - \theta_{a}^{T} D_{t,a}^{T} y_{t,a} + \theta_{a}^{T} D_{t,a}^{T} D_{t,a} \theta_a))
\end{align*}
By definition, we also have
\begin{align*}
p_{t,a}(\theta_a) &\varpropto \text{exp}(-\frac{1}{2}(\theta_a - \hat{\theta_{a}})^{T} A_{t,a} (\theta_a - \hat{\theta_a})) \\
&\varpropto \text{exp}(-\frac{1}{2}(\theta_a^T A_{t,a} \theta_a - \theta_a^T A_{t,a} \hat{\theta_a} - \hat{\theta_a}^T A_{t,a} \theta_a + \hat{\theta}^T A_{t,a} \hat{\theta}))
\end{align*}
And so we have
\begin{align*}
\text{exp}(-\frac{1}{2} (\frac{1}{\lambda} \theta_{a}^{T} \theta_a + y_{t,a}^{T} y_{t,a} - y_{t,a}^{T} D_{t,a} \theta_a - \theta_{a}^{T} D_{t,a}^{T} y_{t,a} + \theta_{a}^{T} D_{t,a}^{T} D_{t,a} \theta_a)) &= \text{exp}(-\frac{1}{2}(\theta_a^T A_{t,a} \theta_a - \theta_a^T A_{t,a} \hat{\theta_a} - \hat{\theta_a}^T A_{t,a} \theta_a + \hat{\theta}^T A_{t,a} \hat{\theta})) \\
\frac{1}{\lambda} \theta_{a}^{T} \theta_a + y_{t,a}^{T} y_{t,a} - y_{t,a}^{T} D_{t,a} \theta_a - \theta_{a}^{T} D_{t,a}^{T} y_{t,a} + \theta_{a}^{T} D_{t,a}^{T} D_{t,a} \theta_a &= \theta_a^T A_{t,a} \theta_a - \theta_a^T A_{t,a} \hat{\theta_a} - \hat{\theta_a}^T A_{t,a} \theta_a + \hat{\theta}^T A_{t,a} \hat{\theta}
\end{align*}
Equating the "second coefficient" of $\theta_a$
$$\theta_a^{T} (\frac{1}{\lambda} I_d)\theta_a + \theta_a^T D_{t,a}^T D_{t,a} \theta_a = \theta_a^T A_{t,a} \theta_a$$
Hence $A_{t,a} = D_{t,a}^T D_{t,a} + \frac{1}{\lambda} I_d$,
Similarly, equating the "first coefficient" of $\theta_a$, we have $$\hat{\theta_a}^T = A_{t,a}^{-1} b_{t,a}$$
