Marginalization of a paramter in Gaussian

Question

If $\theta \sim N(\mu,\sigma_o^2)$ and $\mu \sim N(0, \sigma_1^2)$ what is the marginalized $P(\theta)$.

Is it $N(0,\sigma_o^2+\sigma_1^2)$?

Answer 1

It is often seen that someone writes things like "$\theta\sim N(\mu,\sigma_0^2)$ and $\mu\sim N(0,\sigma_1^2)$" when they ought to write "$\theta\mid\mu\sim N(\mu,\sigma_0^2)$ and $\mu\sim N(0,\sigma_1^2)$", i.e. the conditional distribution of $\theta$ given $\mu$ is that normal distribution.

Now think about the conditional distribution of $\theta-\mu$ given the value of $\mu$. When we're conditioning on $\mu$, we treat $\mu$ as a constant, so we get $(\theta-\mu)\mid\mu \sim N(0,\sigma_0^2)$. Now notice that the conditional distribution of $\theta-\mu$ given $\mu$ actually does not depend on $\mu$. That implies that $\theta-\mu$ is independent of $\mu$. It also implies that the marginal distribution of $\theta-\mu$ is the same as that conditional distribution. So $\theta$ is the sum of two independent normally distibuted random variables, $\theta-\mu$ and $\mu$, and you know their distributions. Now procede as you would whenever you want to know the distribution of the sum of two independent normally distibuted random variables whose distributions you know.