Assume there are two potential univariate linear regression models $M1$ and $M2$, both of which have a prior probability of $0.5$.
Under $M1$, $Y=\alpha + \beta X_1 + \epsilon$, while under $M2$,$Y=\alpha + \beta X_2 + \epsilon$. $\alpha$ and $\beta$ are unknown scalars, but have a known prior distribution. I would be most interested in the case of an uninformative prior distribution, but alternatively normal or simplifying assumptions can be made here! $\epsilon$ is normally distributed with mean $0$ and known variance $\sigma^2$.
Now, we observe a dataset with $n$ observations for $Y$, $X_1$ and $X_2$.
Is it possible (perhaps under further simplifying assumptions) to derive a closed form solution for the posterior probabilities of $M1$ or $M2$ being the correct model?