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Realizations from normal distributions with known precision are used to estimate the mean, but the realizations are not always precisely observed. Instead, only a range of the realization is observed.

I have several questions on this. First: is the resulting posterior still normally distributed? Second, what's the mean and variance of the posterior? And in general, will that Bayesian estimator be asymptotically consistent, given we observe intervals and not observations?

Suppose the realizations are $s_i\sim N(T,1/\rho_i)$ (independent), where $T$ is the value to be estimated, and the prior is distributed as $N(s_0,1/\rho_0)$. We have finite means and variances. In the simplest case, $i=2$, where we only have a range for $s_1$ (say $(-\infty,u]$) and the actual realization of $s_2$, the posterior is $$g(\mu|\tilde{s}_1,s_2)\propto g(\mu) g(s_2|\mu)\int_{-\infty}^u g(s_1|\mu) ds_1.$$ Using the normal density and dropping some constant factors, I arrive at $$g(\mu|\tilde{s}_1,s_2)\propto \exp\left(-\frac{1}{2} \rho_2(s_2-\mu)^2\right)\int_{-\infty}^u \exp \left( -\frac{1}{2}\rho_1(s_1-\mu)^2\right) ds_1 \exp\left(-\frac{1}{2} \rho_0(\mu-s_0)^2\right).$$

To investigate whether the posterior is normal (Q1), I tried writing the integral as Riemann sum (grid $L=x_0<x_1<\ldots<x_N=u$) after taking an $L<0$ with arbitrarily large $|L|$ as integration border, $$\approx \lim_{N\to \infty} \exp\left(-\frac{1}{2} \rho_2(s_2-\mu)^2\right) \sum_{j=0}^{N-1} \left[ (x_{j+1}-x_j) \exp \left( -\frac{1}{2}\rho_1(x_j-\mu)^2\right)\right] \exp\left(-\frac{1}{2} \rho_0(\mu-s_0)^2\right).$$ This is where I am stuck. There are theorems for the sum of normally distributed rvs, but I am unable to apply them to this infinite sum. Or is there a better approach than using the sum?

On the question of consistency, I read martingale convergence theorems are often used to show that the posterior converges and is asymptotically correct. But since the last observation $n$ is replaced by an interval at $n+1$ in my model, the defining property of a filtration is not given for these intervals. How, then, can I show whether the posterior is consistent?

Any help is very much appreciated! Thanks.

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