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How would I compute the marginal posterior distribution of $\mu_1$ and $\mu_2$ if the likelihood $(y | \mu_1,\mu_2) \sim N(\mu_1+\mu_2,1)$ and $\mu_i \sim N(0,1)$

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    $\begingroup$ Welcome to MSE. Please include your own thoughts and the effort made thus far, so that people can work with you accordingly. (Please add those in the body of the question instead of commenting.) $\endgroup$ Feb 16 '20 at 9:19
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Your model is not identifaible. since

$$f(y|\mu_1=1,\mu_2=2)=f(y|\mu_1=2,\mu_2=1)$$ but $$(1,2)\neq (2,1)$$

For resolving identifaibility problem, you should work with $\mu=\mu_1+\mu_2$ like $$y|\mu \sim N(\mu,\sigma^2)$$ or make a condition that guarantee identifaibility.

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