 Solution

Conditionally on ${\lambda }_i$, the proposed precision gives ${\theta }_i\mid {\lambda }_i\sim N(\mu ,4{\tau }^2/{\lambda }_i)$. Thus, with independent $Z\sim N(0,1)$ and ${\lambda }_i\sim {\chi }_4^2$,By the Student t-distribution construction, $\boxed{({\theta }_i-\mu )/\tau \sim t_4}$. Here $\tau$ is a scale rather than the standard deviation: the prior variance of ${\theta }_i$ is $2{\tau }^2$.