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.