2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ This looks like a standard interval-censoring problem; the likelihood should be reasonably easy to deal with. What do you mean by "approximate normally" there? Were you after a Gaussian approximation to the posterior? $\endgroup$ – Glen_b -Reinstate Monica Jun 25 '18 at 5:08
  • $\begingroup$ Thanks for reference! My goal to have an approximate formula for posterior parameters. In this case I will have no need to do any MCMC simulations, just the exact math. As a proposition I want it to be gaussian. $\endgroup$ – Piotr Semenov Jun 25 '18 at 7:25
  • $\begingroup$ I'm sorry, I am unsure of the intended meaning of your comment. $\endgroup$ – Glen_b -Reinstate Monica Jun 25 '18 at 7:48
2
$\begingroup$

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 $\sigma$.

$\endgroup$
  • $\begingroup$ Thanks for your great response! I want the posterior to have a normal distribution. And so my question is how to approximate its mean and variance. $\endgroup$ – Piotr Semenov Jun 24 '18 at 20:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.