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Suppose we have a Normal r.v

$$ x \sim \mathcal{N}(\mu, \sigma^2) $$

and a Normal prior of $\mu$

$$ \mu \sim \mathcal{N}(\theta, \delta^2) $$

I know how to do the Bayesian update with a observation $x_0$:

$$ p(x | x_0) \propto p(x_0 | x) p(x | \mu) p(\mu) $$

However I want to know how to do the update with a bound observation (or how to select the conjugate prior):

$$ p(x | b_1 < x_0 < b_2) $$

Because if we keep Normal dist assumption, it leads to

$$ p(x | b_1 < x_0 < b_2) \propto p(b_1 < x_0 < b_2 | x) p(x|\mu) p(\mu) $$

which is no longer Normal. Thanks for your help.

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