# Conjugate prior for a Gaussian model with shifted variance

Consider a set of observations $\{ y_i \}$ and assume a Gaussian model for these data: $y_i \sim \mathcal{N}(\mu, \sigma^2)$. Suppose the mean parameter $\mu$ is known, but the variance parameter $\sigma^2$ is not. The conjugate prior for $\sigma^2$ is then the scaled inverse chi-squared distribution; consequently, we have a nice posterior. Now, suppose instead of $\mathcal{N}(\mu, \sigma^2)$ our model is $\mathcal{N}(\mu, \sigma^2 + \sigma_0^2)$ where $\sigma_0^2$ is known. What can we say regarding the prior and posterior distributions of $\sigma^2$ in this case?

Regards, Ivan

• Equivalently the model is still ${\cal N}(\mu,\sigma^2)$ but with a prior for $\sigma^2$ having support contained in $[\sigma_0^2,\infty[$. If you take an inverse Gamma prior truncated to $[\sigma_0^2,\infty[$ then you get a truncated inverse Gamma posterior with the same conjugacy relation than the case when there's no truncation. – Stéphane Laurent Jan 26 '13 at 21:15