Posterior for parameters of quantified normal distribution?

Question

Let $X \sim N(\mu, \sigma)$ be a uni-dimensional normal variable with the parameters $\mu$, $\sigma$. Given $n$ fixed cut-offs $-\infty < r_0 < \ldots < r_{n-1} < \infty$, I quantify the variable $X$ with the variable $Q$ defined below: $Q = \begin{cases} 0,~\text{if $X \leq r_0$} \\ \ldots \\ i,~\text{if $r_{i-1} < X \leq r_i$} \\ \ldots \\ n,~\text{if $r_{n-1} < X$} \end{cases}$

Consider we have the sample $q$ drawn from $Q$. Let the prior $p(\mu, \sigma)$ to be just Gaussian. I want to get a (possibly approximate) formula for the posterior $p(\mu, \sigma | q)$. E.g. to approximate it with Gaussian $p(\mu, \sigma | q) \sim N(\mu', \sigma')$ with formulas for $\mu'$ and $\sigma'$. Is it possible?

Answer 1

I don't know what "approximate normally" means in this context. For what it's worth, here is the approach I would take.

Let $\Phi(\,\cdot\,)$ denote the cdf for the standard normal distribution. For $j \in \{0, 1, \ldots, n\}$, define \begin{equation} \pi_j(\mu,\sigma) = \Phi\left(\frac{r_j-\mu}{\sigma}\right) - \Phi\left(\frac{r_{j-1}-\mu}{\sigma}\right) , \end{equation} where $r_{-1} = -\infty$ and $r_n = \infty$. Note $\sum_{j=0}^n \pi_j(\mu,\sigma) = 1$. The discrete random variable $q_i$ is categorical. The probability that $q_i = j$ is given by \begin{equation} p(q_i = j|\mu,\sigma) = \pi_j(\mu,\sigma) . \end{equation} The likelihood for $q_{1:T} = (q_1, \ldots, q_T)$ is \begin{equation} p(q_{1:T}|\mu,\sigma) = \prod_{i=1}^T p(q_i|\mu,\sigma) . \end{equation} The posterior distribution is characterized by \begin{equation} p(\mu,\sigma|q_{1:T}) \propto p(q_{1:T}|\mu,\sigma)\,p(\mu,\sigma) , \end{equation} where $p(\mu,\sigma)$ is the prior for $\mu$ and $\sigma$.