# Bayesian prior and posterior computation for a truncated normal

I have to deal with data in a Bayesian framework, ultimately devising a Gibbs sampler for inferring all my distributions parameters. Specifically, suppose I observe some univariate data distributed according to a truncated Normal in $[a, b]$:

$$X\sim \mathcal{T}\mathcal{N}(\mu,\sigma^{2}, a, b)=\frac{\mathcal{N}(\mu,\sigma^{2})}{\boldsymbol\Phi(\{b-\mu\}/\sigma)-\boldsymbol\Phi(\{a-\mu\}/\sigma)}$$

with parameters $\mu$ and $\sigma^{2}$ for the mean and variance, respectively and $\boldsymbol\Phi(\cdot)$ being the Normal CDF. Now I would like to impose a prior on these parameters, and then devise the form for the posterior $p(\mu, \sigma^{2}|\mathbf{X})$, such that I can derive a full conditional in my Gibbs sampler.

I do not think I could exploit conjugacy here (since the truncated normal should not be in the exponential family), but any closed-form computation for the posterior would be useful. Any ideas on how to choose the prior and obtain a closed form posterior from which it is known how to sample from?

• Please defined completely the truncated Normal distribution as I do not think there is a well-established definition. In particular, (a) what are the truncation points? (b) are $\mu$ and $\sigma$ the parameters of the original normal or the mean and variance of the truncated distribution? As a general comment, indeed there is no conjugate prior and I am pessimistic about a closed form posterior. – Xi'an Apr 20 '18 at 14:30
• you are right, I updated the question. But really, any formalization of the truncated normal would be fine for me – rano Apr 20 '18 at 16:00