I'll trying to understand an exercise where it defines the prior as "$\theta$ is $N(\mu, 1)$ or $N(-\mu, 1)$ with equal probability". I try to add up normal distributions but it will result in a zero mean distribution which is not the expected. Does anyone knows how to write such prior?


Let $(X|θ) ∼ N(θ, 1)$ be observed. Suppose that your prior is such that $θ$ is $N(µ, 1)$ or $N(−µ, 1)$ with equal probabilities. Write the prior distribution and find the posterior after observing $X = x$. Show that

$\mu' = E(\theta|x) = \dfrac{x}{2} + \dfrac{\mu}{2}\dfrac{1 - exp(-\mu x)}{1 + exp(-\mu x)}$


Prior will be:

$p(\theta) = \dfrac{1}{2\sqrt{2\pi}}\exp\lbrace -\frac{1}{2}(\theta - \mu)^2 \rbrace + \dfrac{1}{2\sqrt{2\pi}}\exp\lbrace -\frac{1}{2}(\theta + \mu)^2 \rbrace$


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