= Solution
Write $\overline Y=\beta+\varepsilon$, with independent $\varepsilon\sim N(0,1/n)$ and $\beta\mid H_i\sim N(0,1/q_i)$. The <convolution of independent random variables> is again a <normal distribution>, so the prior predictive laws are
$$
\boxed{\overline Y\mid H_i\sim N(0,V_i),\qquad V_i=1/n+1/q_i.}
$$
These are predictive distributions before observing $y$, hence the <Bayesian model evidence> for the observed mean. The residual information in the original observations is common to both models and cancels in their <Bayes factor>.
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